# An explicit finite difference scheme
for the Camassa-Holm equation

G. M. Coclite
Dipartimento di Matematica
Università degli Studi di Bari
Via E. Orabona 4
70125 Bari, Italy<EMAIL_ADDRESS>,  K. H. Karlsen
Centre of Mathematics for Applications (CMA)
University of Oslo
P.O. Box 1053, Blindern
N–0316 Oslo, Norway<EMAIL_ADDRESS>http://folk.uio.no/kennethk/ and  N.
H. Risebro
Centre of Mathematics for Applications (CMA)
University of Oslo
P.O. Box 1053, Blindern
N–0316 Oslo, Norway<EMAIL_ADDRESS>http://folk.uio.no/nilshr/

###### Abstract.

We put forward and analyze an explicit finite difference scheme for the
Camassa-Holm shallow water equation that can handle general $H^{1}$ initial
data and thus peakon-antipeakon interactions. Assuming a specified condition
restricting the time step in terms of the spatial discretization parameter, we
prove that the difference scheme converges strongly in $H^{1}$ towards a
dissipative weak solution of Camassa-Holm equation.

###### Key words and phrases:

Camassa-Holm equation, hyperbolic-elliptic system, weak solution, finite
difference scheme, convergence

###### 2000 Mathematics Subject Classification:

35G25, 35L05, 65M06, 65M12

This work was supported by the Research Council of Norway through the project
WaveMaker and an Outstanding Young Investigators Award of K. H. Karlsen.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Explicit scheme and main result
  4. 4 Total energy estimate and some consequences
  5. 5 One-sided sup-norm estimate
  6. 6 Higher integrability estimate
  7. 7 Basic convergence results
  8. 8 Strong convergence result
  9. 9 Numerical examples

##  1\. Introduction

In this paper we present and analyze an explicit finite difference scheme for
the Camassa-Holm partial differential equation [7]

(1.1)
$\partial_{t}u-\partial_{txx}^{3}u+3u\partial_{x}u=2\partial_{x}u\partial_{xx}^{2}u+u\partial_{xxx}^{3}u,\qquad(t,x)\in(0,T)\times\mathbb{R},$

which we augment with an initial condition:

(1.2) $u|_{t=0}=u_{0},\qquad u_{0}\in H^{1}(\mathbb{R}),u_{0}\neq 0.$

Rewriting equation (1.1) as

$(1-\partial_{xx}^{2})\left[\partial_{t}u+u\partial_{x}u\right]+\partial_{x}\left(u^{2}+\frac{1}{2}(\partial_{x}u)^{2}\right)=0,$

we see that (for smooth solutions) (1.1) is equivalent to the elliptic-
hyperbolic system

(1.3)
$\partial_{t}u+u\partial_{x}u+\partial_{x}P=0,\qquad-\partial_{xx}^{2}P+P=u^{2}+\frac{1}{2}(\partial_{x}u)^{2}.$

Recalling that $e^{-|x|}/2$ is the Green’s function of the operator
$1-\partial_{xx}^{2}$, (1.3) can be written as

(1.4) $\partial_{t}u+\partial_{x}F(u,\partial_{x}u)=0,\quad
F(u,\partial_{x}u)=\frac{1}{2}\left[u^{2}+e^{-\left|x\right|}\star\left(u^{2}+\frac{1}{2}(\partial_{x}u)^{2}\right)\right],$

which can be viewed as a conservation law with nonlocal flux function. In this
paper the relevant formulation of the Camassa-Holm equation (1.1) is the one
provided by the hyperbolic-elliptic system (1.3) or (1.4).

The Camassa-Holm equation can be viewed as a model for the propagation of
unidirectional shallow water waves [7, 32]; it is a member of the class of
weakly nonlinear and weakly dispersive shallow water models, a class which
already contains the Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM)
equations. In another interpretation the Camassa-Holm equation models finite
length, small-amplitude radial deformation waves in cylindrical compressible
hyperelastic rods [21]. It arises also in the context of differential geometry
as an equation for geodesics of the $H^{1}$-metric on the diffeomorphism
group, see for example [17, 18, 30, 36]. The Camassa-Holm equation possesses
several striking properties such as an inifinite number of conserved
integrals, a bi-Hamiltonian structure, and complete integrability [2, 7, 19,
14, 26]. Moreover, it enjoys an infinite number of non-smooth solitary wave
solutions, called peakons, which are weak solutions of (1.4).

From a mathematical point of view the Camassa-Holm equation has by now become
rather well-studied. While it is impossible to give a complete overview of the
mathematical literature, we shall here mention a few typical results, starting
with the local(-in-time) existence results in [15, 34, 37] and those using
Besov spaces in [23, 22]. It is well-known that global solutions do not exist
and wave-breaking occurs [7]. Wave-breaking means that the solution itself
stays bounded while the spatial derivative becomes unbounded in finite time.

In view of what we have said so far (peakon solutions/wave-breaking) it is
clear that a theory based on weak solutions is essential. In the literature
there are a number of results on (dissipative and conservative) weak solutions
of the Camassa-Holm equation, see [3, 4, 5, 16, 11, 20, 29, 39, 40] and the
references cited therein. In this paper we are interested specifically in the
class of dissipative weak solutions studied by Xin and Zhang [39, 40]. Their
results show, among other things, that there exists a global dissipative weak
solution of (1.1)-(1.2) for any $H^{1}$ initial data $u_{0}$ (peakon-
antipeakon interactions are covered). These solutions are global in the sense
that they are defined past the blow-up time (wave-breaking). More precisely,
suppose $u_{0}\in H^{1}(\mathbb{R})$. Then there exists a global weak
(distributional) solution $u\in L^{\infty}(0,T;H^{1}(\mathbb{R}))$ of (1.1)
satisfying the following properties:
$t\mapsto\left\|u(t,\cdot)\right\|_{H^{1}(\mathbb{R})}$ is non-increasing;
$\partial_{x}u\in L^{p}_{\mathrm{loc}}(\mathbb{R}_{+}\times\mathbb{R})$,
$p<3$;

(1.5)
$\partial_{x}u(t,x)\leq\frac{2}{t}+C\left\|u_{0}\right\|_{H^{1}(\mathbb{R})},\quad\text{for}\quad
t>0,$

for some positive constant $C$. This last item presumably singles out a unique
weak solution. As an example of how this may work we consider the “peakon-
antipeakon” solution given by

(1.6)
$u(t,x)=\tanh(t-1)\left(e^{-\left|x-y(t-1)\right|}-e^{-\left|x+y(t-1)\right|}\right),\quad
y(t)=\log\left(\cosh(t)\right).$

This formula represents a peakon ($e^{-\left|x+y\right|}$) colliding with an
antipeakon ($-e^{-\left|x-y\right|}$) at $x=0$ and $t=1$. Note that
$u(1,x)=0$. How this solution is extended to $t>1$ depends on which solution
concept we adopt. If we use the formula (1.6) also for $t>1$ we get the
conservative solution for which
$\left\|u(t,\cdot)\right\|_{H^{1}(\mathbb{R})}$ is constant for almost all
$t$. We can also extend the solution by defining $u(t,x)=0$ for $t>1$.
Obviously, the “entropy condition” (1.5) will only be satisfied for this
dissipative solution.

Let us now turn to the topic of the present paper, which is the design and
analysis of numerical schemes. The first numerical results for the Camassa-
Holm equation are presented in [8] using a pseudo-spectral scheme. Numerical
simulations with pesudo-spectral schemes are also reported in [25, 31].
Numerical schemes based on multipeakons (thereby exploiting the Hamiltonian
structure of the Camassa-Holm equation) are examined in [6, 9, 10]. In [28],
the authors prove that the multipeakon algorithm from [9, 10] converges to the
solution of the Camassa-Holm equation as the number of peakons tends to
infinity. This convergence result applies to the specific situation where the
initial function $u_{0}\in H^{1}$ is such that $(1-\partial_{xx}^{2})u_{0}$ is
a positive measure. For the same class of initial data, in [27] the authors
prove that a semi-discrete finite difference scheme based on the variable
$m:=(1-\partial_{xx}^{2})u$ converges strongly in $H^{1}$ to the weak solution
identified in [16, 20]. In [33], the authors establish error estimates for a
spectral projection scheme for smooth solutions. In a different direction, an
adaptive high-resolution finite volume scheme is developed and used in [1].
The local discontinuous Galerkin method is adapted to the Camassa-Holm
equation in [41]. Although this work does not provide a rigorous convergence
result for general (non-smooth) solutions, they show that the discrete total
energy is nonincreasing in time, thereby suggesting that the approximate
solutions are of dissipative nature. Besides, they establish an error estimate
for smooth solutions. Finally, multi-symplectic schemes possessing good
conservative properties are suggested and demonstrated in the recent work
[13].

It seems rather difficult to construct numerical schemes for which one can
prove the convergence to a (non-smooth) solution of the Camassa-Holm equation.
This statement is particularly accurate in the case of general $H^{1}$ initial
data and peakon-antipeakon interactions. Indeed, in this context we are only
aware of the recent work [12] in which we prove convergence of a tailored
semi-discrete difference scheme to a dissipative weak solution. Before we can
outline this scheme, let us discretize the spatial domain $\mathbb{R}$ by
specifying the mesh points $x_{j}=j{\Delta x}$, $x_{j+1/2}=(j+1/2){\Delta x}$,
$j=0,\pm 1,\pm 2,\dots$, where ${\Delta x}>0$ is the length between two
consecutive mesh points (the spatial discretization parameter). Let $D_{-}$,
$D$, and $D_{+}$ denote the corresponding backward, central, and forward
difference operators, respectively. The scheme proposed in [12], which is
based on the formulation (1.3), reads

(1.7) $\begin{split}&\frac{d}{dt}u_{j+1/2}+\left(u_{j+1/2}\vee
0\right)D_{-}u_{j+1/2}+\left(u_{j+1/2}\wedge
0\right)D_{+}u_{j+1/2}+D_{+}P_{j}=0,\\\
&-D_{-}D_{+}P_{j}+P_{j}=\left(u_{j+1/2}\vee 0\right)^{2}+\left(u_{j-1/2}\wedge
0\right)^{2}+\frac{1}{2}\left(D_{-}u_{j+1/2}\right)^{2},\end{split}$

where

$u_{j+1/2}(t)\approx u(t,x_{j+1/2}),\quad P_{j}(t)\approx
P(t,x_{j}),\quad\text{for $t\geq 0$ and $j\in\mathbb{Z}$}.$

If we interpret the Camassa-Holm equation (1.4) as a “perturbation” of the
inviscid Burgers equation, then the $u$-part of (1.7) might not come across as
a reasonable (upwind) difference scheme. On the other hand, as pointed out in
[12], the key point is that with (1.7) the quantity $q_{j}:=D_{-}u_{j+1/2}$
satisfies a difference scheme which contains proper upwinding of the transport
term in the equation for $q:=\partial_{x}u$, which reads
$\partial_{t}q+u\partial_{x}q+\frac{q^{2}}{2}+P-u^{2}=0$. Consequently, as is
proved in [12], the scheme (1.7) satisfies a total energy inequality, in which
only the $q$-part of the total energy is dissipated (not the $u$-part, which
is after all continuous). This is the essential starting point for the entire
convergence analysis in [12].

The “semi-discrete” equation in (1.7) constitutes an infinite system of
ordinary differential equations which must be solved by some numerical method.
The main purpose of the present paper is to show that a fully discrete version
of the scheme used in [12] produces a convergent sequence of approximate
solutions, and that the limit is a dissipative weak solution to (1.1). The
fully discrete version that is analyzed in this paper is based on replacing
the time derivative in (1.7) by a forward difference, i.e.,

$u_{j+1/2}^{\prime}(t)\to
D_{+}^{t}u_{j+1/2}^{n}:=\frac{u^{n+1}_{j+1/2}-u_{j+1/2}^{n}}{{\Delta t}},$

and evaluating the rest of (1.7) at $t^{n}:=n{\Delta t}$. Now $u_{j+1/2}^{n}$
should approximate the exact solution $u$ at the point $(t^{n},x_{j+1/2})$.
This gives the fully discrete scheme

(1.8) $\begin{split}D_{+}^{t}u_{j+1/2}^{n}+\left(u_{j+1/2}^{n}\vee
0\right)D_{-}u_{j+1/2}^{n}+\left(u_{j+1/2}^{n}\wedge
0\right)D_{+}u_{j+1/2}^{n}+D_{+}P_{j}^{n}=0\\\
-D_{-}D_{+}P_{j}^{n}+P_{j}^{n}=\left(u_{j-1/2}^{n}\vee
0\right)^{2}+\left(u_{j+1/2}^{n}\wedge
0\right)^{2}+\frac{1}{2}\left(D_{-}u_{j+1/2}^{n}\right)^{2},\end{split}$

where $P_{j}^{n}$ approximates $P(t^{n},x_{j})$. As in [12] this is a
difference scheme which is tailored so that it gives an upwind scheme for the
equation satisfied by $q:=\partial_{x}u$.

The main aim of this paper is prove that the fully discrete (explicit) scheme
(1.8) converges to a dissipative weak solution of the Camassa-Holm equation.
The starting point of the analysis is a total energy estimate, showing that
the $H^{1}$ norm of the approximate solutions is (almost) nonincreasing in
time. To this end, we must assume that

(1.9) ${\Delta t}=\mathcal{O}({\Delta x}^{2}\log(1+{\Delta x}^{\theta}))$

for some $\theta>0$ as ${\Delta x}\to 0$. This is a very severe condition, and
it may seem that when using this method in practice one should use very small
time steps. However, this is not a Neumann type stability criterion, and we do
not have blow up if it is violated. Indeed, practical experiments indicate
stability and convergence if ${\Delta t}=\mathcal{O}({\Delta x})$.

By appropriately extending the difference solution (1.8) to a function
$u_{{\Delta x}}(t,x)$ defined at all points $(t,x)$ in the domain, we prove
under condition (1.9) that $\left\\{u_{{\Delta x}}\right\\}_{{\Delta x}>0}$
converges strongly in $H^{1}$ to a dissipative weak solution of the Camassa-
Holm equation (1.1)-(1.2). Regarding the proof, we adapt the “renormalization”
approach used in [12] for the semi-discrete scheme, but there are several
essential deviations and many parts of the convergence proof are substantially
more involved and/or different. These differences are mainly due to the fact
that the semi-discrete scheme, when viewed as a fully discrete scheme with
“infinitely small time steps”, has a large and stabilizing numerical
viscosity. Regarding the fully discrete (explict) scheme (1.8), to account for
this lack of numerical viscosity the convergence analysis relies heavily on
the CFL condition (1.9) and differennt With reference to the differences
between the semi-discrete and fully discrete schemes, let us here point out
just one aspect, namely that the $H^{1}$ norm of the fully discrete
approximation is not entirely nonincreasing but can grow slightly with a
growth factor that, however, tends to zero as ${\Delta x}\to 0$. Compared to
semi-implcit case [12], the proof is notably more complicated and involves
working with a version of the scheme (1.8) in which the quadratic terms have
been suitably truncated.

The paper is organized as follows: In Section 2 we introduce some notation to
improve the readability and recall a few mathematical results relevant for the
convergence analysis. The finite difference scheme and its convergence theorem
are stated in Section 3. The convergence theorem is a consequence of the
results proved in Sections 4-8. Finally, we present a numerical example in
Section 9.

Throughout this paper we use $C$ to denote a generic constant; the actual
value of $C$ may change from one line to the next in a calcuation. We also use
the notation that $a_{i}\lesssim b_{i}$ to mean that $a_{i}\leq Cb_{i}$ for
some positive constant $C$ which is independent of $i$.

##  2\. Preliminaries

In what follows, ${\Delta x}$ and ${\Delta t}$ denote two small positive
numbers. Unless otherwise stated, the indices $j$ and $n$ will run over
$\mathbb{Z}$ and $0,\ldots N$, respectively, where $N{\Delta t}=T$ for a fixed
final time $T>0$. For such indices we set $x_{j}=j{\Delta x}$,
$x_{j+1/2}=(j+1/2){\Delta x}$, $t^{n}=n{\Delta t}$, and introduce the grid
cells

$I_{j}=[x_{j-1/2},x_{j+1/2}),\quad I^{n}=[t^{n},t^{n+1}),\quad\text{and}\quad
I^{n}_{j}=I_{j}\times I^{n}.$

The following notations will be used frequently:

$a\vee 0=\max\left\\{a,0\right\\}=\frac{a+\left|a\right|}{2},\qquad a\wedge
0=\min\left\\{a,0\right\\}=\frac{a-\left|a\right|}{2}.$

For $n\in\left\\{0,\dots,N\right\\}$, let
$v^{n}=\left\\{v_{j}^{n}\right\\}_{j\in\mathbb{Z}}$ denote an arbitrary
sequence, where $n$ refers to “time” and $j$ to “space”. We will frequently
employ the following finite difference operators:

$\begin{gathered}D_{+}v^{n}_{j}:=\frac{v^{n}_{j+1}-v^{n}_{j}}{{\Delta
x}},\qquad D_{-}v^{n}_{j}:=\frac{v^{n}_{j}-v^{n}_{j-1}}{{\Delta x}},\\\
Dv^{n}_{j}:=\frac{D_{+}v^{n}_{j}+D_{-}v^{n}_{j}}{2}=\frac{v^{n}_{j+1}-v^{n}_{j-1}}{2{\Delta
x}}\quad\text{and}\quad
D_{+}^{t}v^{n}_{j}=\frac{v^{n+1}_{j}-v^{n}_{j}}{{\Delta t}}.\end{gathered}$

We also use the notations

$\displaystyle\left\|v^{n}\right\|_{\ell^{p}}$ $\displaystyle:=\left(\Delta
x\sum\limits_{j\in\mathbb{Z}}\left|v^{n}_{j}\right|^{p}\right)^{\frac{1}{p}},\,1\leq
p<\infty,\qquad\left\|v^{n}\right\|_{\ell^{\infty}}:=\sup_{j}\left|v_{j}^{n}\right|,$
$\displaystyle\left\|v^{n}\right\|_{h^{1}}$ $\displaystyle:=\left({\Delta
x}\sum\limits_{j\in\mathbb{Z}}\left[\left(v^{n}_{j}\right)^{2}+(D_{-}v^{n}_{j})^{2}\right]\right)^{\frac{1}{2}}.$

Occasionally, we also use the “space-time” $\ell^{p}$ norms of
$v=\left\\{v^{n}\right\\}_{n=0}^{N}=\left\\{v_{j}^{n}\right\\}_{j,n}$:

$\left\|v\right\|_{\ell^{p}}:=\left({\Delta
t}\sum_{n=0}^{N}\left\|v^{n}\right\|_{\ell^{p}}^{p}\right)^{1/p}.$

Note that if $v\in\ell^{p}$, $p<\infty$, then $\lim_{j\to\pm\infty}v_{j}=0$.

Let $\left\\{v_{j}\right\\}_{j\in\mathbb{Z}}$ and
$\left\\{w_{j}\right\\}_{j\in\mathbb{Z}}$ denote two arbitrary (spatial)
sequences. Suppose $\left\|\left\\{v_{j}\right\\}_{j}\right\|_{h^{1}}<\infty$.
Then the following discrete Sobolev inequality holds:

(2.1)
$\left\|\left\\{v_{j}\right\\}_{j}\right\|_{\ell^{\infty}}\leq\frac{1}{\sqrt{2}}\left\|\left\\{v_{j}\right\\}_{j}\right\|_{h^{1}}.$

The discrete product rule takes the form

(2.2) $D_{\pm}\left(v_{j}w_{j}\right)=v_{j}D_{\pm}w_{j}+D_{\pm}v_{j}w_{j\pm
1}.$

Moreover, the discrete chain rule states

(2.3) $D_{\pm}f(v_{j})=f^{\prime}(v_{j})D_{\pm}v_{j}\pm\frac{{\Delta
x}}{2}f^{\prime\prime}(\xi_{j}^{\pm})(D_{\pm}v_{j})^{2},\qquad f\in C^{2},$

for some number $\xi_{j}^{\pm}$ between $v_{j\pm 1}$ and $v_{j}$.

We continue to collect some handy results for later use, starting with a
discrete Gronwall inequality.

######  Lemma 2.1.

Assume that $c^{k}\geq 0$ and $f^{k}\geq 0$ for all $k=0,\dots,N$, and that
the sequence $\left\\{u^{n}\right\\}_{n=0}^{N}$ satisfies the difference
inequality

(2.4) $D_{+}^{t}u^{n}+f^{n}\leq c^{n}u^{n},\qquad n=0,\dots,N-1.$

If $u^{n}\geq 0$ for all $n=0,\dots,N$, then

$u^{N}+\exp\left({\Delta t}\sum_{n=0}^{N-1}c^{n}\right){\Delta
t}\sum_{n=0}^{N-1}\exp\left(-{\Delta
t}\sum_{k=0}^{n}c^{k}\right)f^{n}\leq\exp\left({\Delta
t}\sum_{n=0}^{N-1}c^{n}\right)u^{0}.$

###### Proof.

Set $R^{n}=\exp\left(-{\Delta t}\sum_{k=0}^{n-1}c^{k}\right)$. Then we have

$\displaystyle D_{+}^{t}R^{n}$ $\displaystyle=\frac{1}{{\Delta
t}}\left(R^{n+1}-R^{n}\right)$ $\displaystyle=\exp\left(-{\Delta
t}\sum_{k=0}^{n}c^{k}\right)\frac{1}{{\Delta t}}\left(1-\exp\left({\Delta
t}c^{n}\right)\right)$ $\displaystyle\leq R^{n+1}\frac{1}{{\Delta
t}}\left(1-(1+c^{n}{\Delta t})\right)=-c^{n}R^{n+1}.$

Hence, multiplying (2.4) with $R^{n+1}$ we arrive at

$D_{+}^{t}\left(R^{n}u^{n}\right)=D_{+}^{t}u^{n}R^{n+1}+D_{+}^{t}R^{n}u^{n}\leq-f^{n}R^{n+1}.$

Multiplying this by ${\Delta t}$ and summing over $n$, we see that the lemma
holds. ∎

The next lemma contains estimates for the solution of a discrete version of
the differential equation $P-\partial_{xx}^{2}P=f$.

######  Lemma 2.2.

Let $\left\\{f_{j}\right\\}_{j\in\mathbb{Z}}$ be a sequence in
$\ell^{1}\cap\ell^{2}$, and denote by
$\left\\{P_{j}\right\\}_{j\in\mathbb{Z}}$ the solution to the difference
equation

(2.5) $P_{j}-D_{-}D_{+}P_{j}=f_{j},\qquad j\in\mathbb{Z}.$

Introducing the notations

$h=\left(1+2\frac{1-e^{-\kappa}}{({\Delta
x})^{2}}\right)^{-1},\quad\kappa=\ln\left(1+\frac{{\Delta
x}^{2}}{2}+\frac{{\Delta x}}{2}\sqrt{4+{\Delta x}^{2}}\right),$

the solution $\left\\{P_{j}\right\\}_{j\in\mathbb{Z}}$ takes the form

(2.6)
$P_{j}=h\sum\limits_{i\in\mathbb{Z}}e^{-\kappa\left|j-i\right|}f_{i},\qquad
j\in\mathbb{Z}.$

Moreover, the following estimates hold:

(2.7)
$\displaystyle\left\|\left\\{P_{j}\right\\}_{j}\right\|_{\ell^{\infty}},\left\|\left\\{P_{j}\right\\}_{j}\right\|_{\ell^{1}}$
$\displaystyle\leq C\left\|\left\\{f_{j}\right\\}\right\|_{\ell^{1}},$ (2.8)
$\displaystyle\left\|\left\\{D_{+}P_{j}\right\\}_{j}\right\|_{\ell^{\infty}},\left\|\left\\{D_{+}P_{j}\right\\}_{j}\right\|_{\ell^{1}}$
$\displaystyle\leq C\left\|\left\\{f_{j}\right\\}_{j}\right\|_{\ell^{1}},$
(2.9) $\displaystyle\left\|\left\\{P_{j}\right\\}_{j}\right\|_{h^{1}}$
$\displaystyle\leq C\left\|\left\\{f_{j}\right\\}_{j}\right\|_{\ell^{2}},$

where $C>0$ is a constant independent of ${\Delta x}$.

###### Proof.

To verify the solution formula (2.6), we define $p_{i}$ by

$p_{i}=ce^{-\sigma\left|i\right|},$

for some constants $c$ and $\sigma$ yet to be found. We shall choose these so
that

$(I-D_{+}D_{-})p_{i}=\begin{cases}1,&\text{if $i=0$,}\\\
0,&\text{otherwise.}\end{cases}$

If we find that this holds with $\sigma=\kappa$ and $c=h$ then (2.6) holds. We
observe that for $i\neq 0$

$D_{+}D_{-}p_{i}=ce^{-\sigma\left|i\right|}2\frac{\cosh(\sigma)-1}{{\Delta
x}^{2}}.$

Hence $\sigma$ must satisfy

$\sigma=\cosh^{-1}\left(1+\frac{{\Delta x}^{2}}{2}\right)=\kappa.$

For $i=0$ we find that

$p_{0}-D_{+}D_{-}p_{0}=c\left(1-\frac{2}{{\Delta
x}^{2}}\left(e^{-\kappa}-1\right)\right).$

If this is to be equal $1$ then $c=h$.

For later use, one should observe that

(2.10) $h=\frac{{\Delta x}}{2}+{\mathcal{O}}({\Delta
x}^{2}),\quad\frac{\left|e^{\kappa}-1\right|}{{\Delta
x}}=1+\mathcal{O}({\Delta x}),\quad\frac{\left|e^{-\kappa}-1\right|}{{\Delta
x}}=1+\mathcal{O}({\Delta x}).$

For any $j\in\mathbb{Z}$, we have

$\left|P_{j}\right|\lesssim\left\|\left\\{f_{j}\right\\}\right\|_{\ell^{1}}$

Furthermore,

$\displaystyle\left\|\left\\{P_{j}\right\\}\right\|_{\ell^{1}}\leq
2h\sum\limits_{i}\bigg{[}\Delta
x\sum\limits_{j}e^{-\kappa\left|j-i\right|}\bigg{]}\left|f_{i}\right|\lesssim\left\|\left\\{f_{i}\right\\}\right\|_{\ell^{1}},$

Hence, we have proved (2.7).

From (2.6),

$\displaystyle D_{+}P_{j}$ $\displaystyle=\frac{P_{j+1}-P_{j}}{{\Delta x}}$
$\displaystyle=h\sum\limits_{i}\frac{e^{-\kappa\left|i-j-1\right|}-e^{-\kappa\left|i-j\right|}}{{\Delta
x}}f_{i}$
$\displaystyle=h\sum\limits_{i=j}^{\infty}\frac{e^{-\kappa(i-j-1)}-e^{-\kappa(i-j)}}{{\Delta
x}}f_{i}+h\sum\limits_{i=-\infty}^{j-1}\frac{e^{\kappa(i-j-1)}-e^{\kappa(i-j)}}{{\Delta
x}}f_{i}$
$\displaystyle=h\sum\limits_{i=j}^{\infty}e^{-\kappa(i-j)}\frac{e^{\kappa}-1}{{\Delta
x}}f_{i}+h\sum\limits_{i=-\infty}^{j-1}e^{\kappa(i-j)}\frac{e^{-\kappa}-1}{{\Delta
x}}f_{i}.$

Using (2.10) we acquire from this the following two estimates:

$\left|D_{+}P_{j}\right|\lesssim
h\sum\limits_{i}e^{-\kappa\left|i-j\right|}\left|f_{i}\right|\lesssim\left\|\left\\{f_{i}\right\\}\right\|_{\ell^{1}}$

and

$\left\|\left\\{D_{+}P_{j}\right\\}_{j}\right\|_{\ell^{1}}\lesssim h{\Delta
x}\sum\limits_{j,i}e^{-\kappa\left|i-j\right|}\left|f_{i}\right|\lesssim\left\|\left\\{f_{i}\right\\}\right\|_{\ell^{1}},$

Therefore (2.8) holds.

It remains to prove (2.9). To this end, we multiply the equation (2.5) by
${\Delta x}P_{j}$ and perform a summation by parts to discover

$\displaystyle\left\|\left\\{P_{j}\right\\}_{j}\right\|_{h^{1}}^{2}$
$\displaystyle={\Delta
x}\sum_{j}P_{j}f_{j}\leq\frac{1}{2}\left\|\left\\{P_{j}\right\\}_{j}\right\|_{\ell^{2}}^{2}+\frac{1}{2}\left\|\left\\{f_{j}\right\\}_{j}\right\|_{\ell^{2}}^{2},$

from which (2.9) follows. ∎

We shall routinely use some well-known results related to weak convergence,
which we collect in a lemma (for proofs, see, e.g., [24]). Throughout the
paper we use overbars to denote weak limits.

######  Lemma 2.3.

Let $O$ be a bounded open subset of $\mathbb{R}^{M}$, with $M\geq 1$.

Let $\left\\{v_{n}\right\\}_{n\geq 1}$ be a sequence of measurable functions
on $O$ for which

$\sup_{n\geq 1}\int_{O}\Phi(\left|v_{n}(y)\right|)\,dy<\infty,$

for some given continuous function $\Phi:[0,\infty)\to[0,\infty)$. Then along
a subsequence as $n\to\infty$

$g(v_{n})\rightharpoonup\overline{g(v)}$ in $L^{1}(O)$

for all continuous functions $g:\mathbb{R}\to\mathbb{R}$ satisfying

$\lim_{\left|v\right|\to\infty}\frac{\left|g(v)\right|}{\Phi(\left|v\right|)}=0.$

Let $g\colon\mathbb{R}\to(-\infty,\infty]$ be a lower semicontinuous convex
function and $\left\\{v_{n}\right\\}_{n\geq 1}$ a sequence of measurable
functions on $O$, for which

$\textit{$v_{n}\rightharpoonup v$ in $L^{1}(O)$, $g(v_{n})\in L^{1}(O)$ for
each $n$, $g(v_{n})\rightharpoonup\overline{g(v)}$ in $L^{1}(O)$}.$

Then

$g(v)\leq\overline{g(v)}$ a.e. on $O$.

Moreover, $g(v)\in L^{1}(O)$ and

$\int_{O}g(v)\,dy\leq\liminf_{n\to\infty}\int_{O}g(v_{n})\,dy.$

If, in addition, $g$ is strictly convex on an open interval
$(a,b)\subset\mathbb{R}$ and

$\text{$g(v)=\overline{g(v)}$ a.e.~{}on $O$},$

then, passing to a subsequence if necessary,

$v_{n}(y)\to v(y)$ for a.e. $y\in\left\\{y\in O\mid v(y)\in(a,b)\right\\}$.

Let $X$ be a Banach space and denote by $X^{\star}$ its dual. The space
$X^{\star}$ equipped with the weak-$\star$ topology is denoted by
$X^{\star}_{\mathrm{weak}}$, while $X$ equipped with the weak topology is
denoted by $X_{\mathrm{weak}}$. By the Banach-Alaoglu theorem, a bounded ball
in $X^{\star}$ is $\sigma(X^{\star},X)$-compact. If $X$ separable, then the
weak-$\star$ topology is metrizable on bounded sets in $X^{\star}$, and thus
one can consider the metric space
$C\left([0,T];X^{\star}_{\mathrm{weak}}\right)$ of functions $v:[0,T]\to
X^{\star}$ that are continuous with respect to the weak topology. We have
$v_{n}\to v$ in $C\left([0,T];X^{\star}_{\mathrm{weak}}\right)$ if $\langle
v_{n}(t),\phi\rangle_{X^{\star},X}\to\langle v(t),\phi\rangle_{X^{\star},X}$
uniformly with respect to $t$, for any $\phi\in X$. The following lemma is a
consequence of the Arzelà-Ascoli theorem:

######  Lemma 2.4.

Let $X$ be a separable Banach space, and suppose $v_{n}\colon[0,T]\to
X^{\star}$, $n=1,2,\dots$, is a sequence of measurable functions such that

$\left\|v_{n}\right\|_{L^{\infty}([0,T];X^{\star})}\leq C,$

for some constant $C$ independent of $n$. Suppose the sequence

$[0,T]\ni t\mapsto\langle v_{n}(t),\Phi\rangle_{X^{\star},X},\quad
n=1,2,\dots,$

is equi-continuous for every $\Phi$ that belongs to a dense subset of $X$.
Then $v_{n}$ belongs to $C\left([0,T];X^{\star}_{\mathrm{weak}}\right)$ for
every $n=1,2,\dots$, and there exists a function $v\in
C\left([0,T];X^{\star}_{\mathrm{weak}}\right)$ such that along a subsequence
as $n\to\infty$

$\text{$v_{n}\to v$ in $C\left([0,T];X^{\star}_{\mathrm{weak}}\right)$}.$

##  3\. Explicit scheme and main result

In this section we present the fully discrete (explicit) difference scheme for
the Cammassa-Holm equation (1.3), which generates sequences
$\left\\{u_{j+1/2}^{n}\right\\}$ and $\left\\{P_{j}^{n}\right\\}$ for
$(n,j)\in\left\\{0,\dots,N\right\\}\times\mathbb{Z}$. We let
$\left\\{u_{j+1/2}^{n}\right\\}$ solve the explicit difference equation

(3.1) $D_{+}^{t}u_{j+1/2}^{n}+\left(u_{j+1/2}^{n}\vee
0\right)D_{-}u_{j+1/2}^{n}+\left(u_{j+1/2}^{n}\wedge
0\right)D_{+}u_{j+1/2}^{n}+D_{+}P_{j}^{n}=0,$

where the initial values are specified as follows:

(3.2) $u_{j+1/2}^{0}=u_{0}(x_{j+1/2}),$

Given $\left\\{u_{j+1/2}^{n}\right\\}$, we determine
$\left\\{P_{j}^{n}\right\\}$ by solving

(3.3) $-D_{-}D_{+}P_{j}^{n}+P_{j}^{n}=\left(u_{j+1/2}^{n}\vee
0\right)^{2}+\left(u_{j-1/2}^{n}\wedge
0\right)^{2}+\frac{1}{2}\left(D_{-}u_{j+1/2}^{n}\right)^{2},$

which is a linear system of equations that can be solved as outlined in Lemma
2.2.

Next, let us derive the difference scheme satisfied by

(3.4) $q_{j}^{n}=D_{-}u_{j+1/2}^{n}.$

This will be done by applying the difference operator $D_{-}$ to the
$u$-equation (3.1). To this end, we apply the discrete product rule to find

$D_{-}\left[\left(u_{j+1/2}^{n}\vee
0\right)D_{-}u_{j+1/2}^{n}\right]=\left(u_{j-1/2}^{n}\vee
0\right)D_{-}q_{j}+D_{-}\left(u_{j+1/2}^{n}\vee 0\right)q_{j}^{n}$

and

$D_{-}\left[\left(u_{j+1/2}^{n}\wedge
0\right)D_{+}u_{j+1/2}^{n}\right]=\left(u_{j+1/2}^{n}\wedge
0\right)D_{+}q_{j}+D_{-}\left(u_{j+1/2}^{n}\wedge 0\right)q_{j},$

so that

(3.5) $\begin{split}&D_{-}\left[\left(u_{j+1/2}^{n}\vee
0\right)D_{-}u_{j+1/2}^{n}+\left(u_{j+1/2}^{n}\wedge
0\right)D_{+}u_{j+1/2}^{n}\right]\\\ &\qquad\qquad=\left(u_{j-1/2}^{n}\vee
0\right)D_{-}q_{j}^{n}+\left(u_{j+1/2}^{n}\wedge
0\right)D_{+}q_{j}^{n}+\left({q_{j}^{n}}\right)^{2}.\end{split}$

The $P$-equation (3.3) rephrased in terms of $q$ reads

(3.6) $-D_{-}D_{+}P_{j}^{n}+P_{j}^{n}=\left(u_{j+1/2}^{n}\vee
0\right)^{2}+\left(u_{j-1/2}^{n}\wedge
0\right)^{2}+\frac{1}{2}\left({q_{j}^{n}}\right)^{2}.$

Employing (3.5) and (3.6) when applying $D_{-}$ to the $u$-equation in (3.1)
yields

(3.7) $\begin{split}D_{+}^{t}q_{j}^{n}&+\left(u_{j-1/2}^{n}\vee
0\right)D_{-}q_{j}^{n}+\left(u_{j+1/2}^{n}\wedge 0\right)D_{+}q_{j}^{n}\\\
&+\frac{\left({q_{j}^{n}}\right)^{2}}{2}+P_{j}^{n}-\left(u_{j+1/2}^{n}\vee
0\right)^{2}-\left(u_{j-1/2}^{n}\wedge 0\right)^{2}=0.\end{split}$

Regarding the initial values, in view of (3.4) and (3.2), we observe that

(3.8) $q_{j}^{0}=\frac{1}{{\Delta
x}}\int_{I_{j}}\partial_{x}u_{0}(x)\,dx,\qquad j\in\mathbb{Z}.$

Since the variable $q=\partial_{x}u$ can be discontinuous, (3.7) represents a
natural upwind discretization of the equation for $q$,
$\partial_{t}q+u\partial_{x}q+\frac{q^{2}}{2}-u^{2}+P=0$.

The main result of this paper is the convergence of the scheme to a
dissipative weak solution of (1.1)-(1.2), which is defined in the following
sense [39, 40]:

######  Definition 3.1.

Fix a final time $T>0$. We call a function
$u\colon[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ a weak solution of the
Cauchy problem for (1.1)-(1.2) on $[0,T]\times\mathbb{R}$ if

  1. (D.1)

$u\in C([0,T]\times\mathbb{R})\cap L^{\infty}(0,T;H^{1}(\mathbb{R}))$;

  2. (D.2)

For all $s$ and $t$ in $[0,T]$, with $s\leq t$, we have
$\left\|u(t,\cdot)\right\|_{H^{1}(\mathbb{R})}\leq\left\|u(s,\cdot)\right\|_{H^{1}(\mathbb{R})}$.

  3. (D.3)

$u$ satisfies (1.3) in the sense of distributions on $(0,T)\times\mathbb{R}$;

  4. (D.4)

$u(0,x)=u_{0}(x)$ for every $x\in\mathbb{R}$;

  5. (D.5)

If, in addition, there exists a positive constant $K$ such that

$u_{x}(t,x)\leq\frac{2}{t}+K\left\|u_{0}\right\|_{H^{1}(\mathbb{R})}^{2},\qquad(t,x)\in(0,T]\times\mathbb{R},$

then we call $u$ a dissipative weak solution of the Cauchy problem
(1.1)-(1.2).

In addition to $\partial_{x}u\in L^{\infty}(0,T;L^{2}(\mathbb{R}))$, cf.
(D.2), the dissipative weak solutions $u$ that we construct in this paper will
possess an improved integrability property, namely $\partial_{x}u\in
L^{p}_{\mathrm{loc}}((0,T)\times\mathbb{R})$ for $p<3$, i.e.,

$\int_{0}^{T}\int_{a}^{b}\left|\partial_{x}u\right|^{p}\,dx\,dt\leq
C(a,b,T,p),\quad\text{$\forall a,b\in\mathbb{R}$, $a<b$.}$

To state our main convergence result and also for later use, we need to
introduce some functions (interpolations of the difference approximations)
that are defined at all points $(t,x)$ in the domain. We begin by defining the
functions

$\left.\begin{aligned}
q_{j}(t)&=q_{j}^{n}+\left(t-t^{n}\right)D_{+}^{t}q_{j}^{n},\\\
u_{j+1/2}(t)&=u_{j+1/2}^{n}+\left(t-t^{n}\right)D_{+}^{t}u_{j+1/2}^{n}\end{aligned}\right\\}\quad\text{for
$t\in I^{n}$.}$

With the aid of these we define

(3.9) $q_{{\Delta x}}(t,x)=q_{j}(t),\qquad(t,x)\in I_{j}^{n}$

and

(3.10) $u_{{\Delta
x}}(t,x)=u_{j-1/2}(t)+\left(x-x_{j-1/2}\right)q_{j}(t),\qquad\text{for
$(t,x)\in I_{j}^{n}$,}$

for $j\in\mathbb{Z}$, $n=0,\dots,N-1$. Note that $t\mapsto u_{{\Delta
x}}(t,x)$ is a continuous function, since $u_{j-1/2}(t)$ and $q_{j}(t)$ are
continuous. Regarding the continuity in $x$ we have that

$\displaystyle\lim_{x\uparrow x_{j+1/2}}u_{{\Delta x}}(t,x)$
$\displaystyle=u_{j-1/2}(t)+\left(x_{x+1/2}-x_{j-1/2}\right)q_{j}(t)$
$\displaystyle=u_{j-1/2}^{n}+\left(t-t^{n}\right)D_{+}^{t}u_{j-1/2}^{n}+{\Delta
x}\left(q_{j}^{n}+\left(t-t^{n}\right)D_{+}^{t}q_{j}^{n}\right)$
$\displaystyle=u_{j-1/2}^{n}+\left(t-t^{n}\right)D_{+}^{t}u_{j-1/2}^{n}+\left(u_{j+1/2}^{n}-u_{j-1/2}^{n}\right)$
$\displaystyle\qquad\quad+\left(t-t^{n}\right)D_{+}^{t}\left(u_{j+1/2}^{n}-u_{j-1/2}^{n}\right)=u_{j+1/2}(t),$

and therefore $u_{{\Delta x}}$ is continuous, and furthermore
$\partial_{x}u_{{\Delta x}}=q_{{\Delta x}}$ almost everywhere. Observe also
that, due to (3.8), there holds $q_{{\Delta x}}(0,x)\to\partial_{x}u_{0}$ in
$L^{2}(\mathbb{R})$ as ${\Delta x}\to 0$. Similarly to $u_{{\Delta x}}$, we
define a function $P_{{\Delta x}}$ by bilinear interpolation. First, let

$P_{j}(t)=P^{n}_{j}+(t-t^{n})D_{+}^{t}P_{j}^{n},\qquad t\in I^{n},$

and then define

(3.11) $P_{{\Delta x}}(t,x)=P_{j}(t)+(x-x_{j})D_{+}P_{j}(t),\qquad(t,x)\in
I_{j}^{n},$

for $j\in\mathbb{Z}$, $n=0,\dots,N-1$.

We are now in a position to state our main result.

######  Theorem 3.1.

Suppose (1.2) holds. Let $\left\\{u_{{\Delta x}}\right\\}_{{\Delta x}>0}$ be a
sequence defined by (3.10) and (3.1)-(3.4). Then, along a subsequence as
${\Delta x}\downarrow 0$,

$\text{$u_{{\Delta x}}\to u$ in
$H_{\mathrm{loc}}^{1}((0,T)\times\mathbb{R})$},$

where $u$ is a dissipative weak solution of the Cauchy problem (1.1)-(1.2).

This theorem is a consequence of the results stated and proved in Sections
4-8.

##  4\. Total energy estimate and some consequences

The purpose of this section is establish a discrete total energy estimate for
the difference scheme (3.1)-(3.4).

######  Lemma 4.1.

Assume that ${\Delta x}$ and ${\Delta t}$ are related through the CFL type
condition

(4.1) ${\Delta t}<\frac{\log(1+{\Delta x}^{\theta})\,{\Delta
x}^{2}}{C\left\|u_{0}\right\|_{H^{1}(\mathbb{R})}^{2}\left(1+{\Delta
x}^{2}\right)},$

where $C$ is a constant (to be detailed in the proof of the lemma) that is
independent of ${\Delta x}$ and $u_{0}$ and $\theta>0$. Then, for any
$N_{0}\in\left\\{0,\dots,N\right\\}$, and for all sufficiently small ${\Delta
x}$,

(4.2) $\left\|u^{N_{0}}\right\|_{h^{1}}^{2}+{\Delta x}^{2}{\Delta
t}\sum_{n=0}^{N_{0}-1}\sum_{j\in\mathbb{Z}}\left|u_{j+1/2}^{n}\right|\left(D_{-}D_{+}u_{j+1/2}^{n}\right)^{2}\leq
e^{t^{N}{\Delta x}^{\theta}}\left\|u^{0}\right\|_{h^{1}}^{2}.$

###### Proof.

For the proof of (4.2), we shall need to introduce an auxiliary difference
scheme. To this end, we start by defining the cut-off function

$f^{M}(u)=\begin{cases}-M,\;&u<M,\\\ u,\;&u\in[-M,M],\\\ +M\;&u>M,\end{cases}$

where $M>0$ is a fixed constant (to be determined later on). Now, let
$\left\\{\tilde{u}_{j+1/2}^{n}\right\\}$ and
$\left\\{\tilde{P}_{j}^{n}\right\\}$ solve the following system of difference
equations:

(4.3) $\displaystyle D_{+}^{t}\tilde{u}_{j+1/2}^{n}$
$\displaystyle+\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)D_{-}\tilde{u}_{j+1/2}^{n}$
$\displaystyle+\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)D_{+}\tilde{u}_{j+1/2}^{n}+D_{+}\tilde{P}_{j}^{n}=0,$

for $n=0,\dots,N-1$ and $j\in\mathbb{Z}$, and

$\displaystyle-D_{-}D_{+}\tilde{P}_{j}^{n}+\tilde{P}_{j}^{n}$
$\displaystyle=\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)\left(\tilde{u}_{j+1/2}^{n}\vee 0\right)$
$\displaystyle\qquad+\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\wedge
0\right)\left(\tilde{u}_{j-1/2}^{n}\wedge 0\right)$
$\displaystyle\qquad+\frac{1}{2}D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right)D_{-}\tilde{u}_{j+1/2}^{n},$

for $n=0,\dots,N$ and $j\in\mathbb{Z}$. Regarding the initial data, we set
$\tilde{u}_{j}^{0}=u_{j}^{0}$ for $j\in\mathbb{Z}$.

If we define $\tilde{q}_{j}^{n}:=D_{-}\tilde{u}_{j+1/2}^{n}$, then it is
straightforward to see that $\left\\{\tilde{q}_{j}^{n}\right\\}$ satisfies the
difference equation

(4.4)
$\begin{split}D_{+}^{t}\tilde{q}_{j}^{n}&+\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\vee
0\right)D_{-}\tilde{q}_{j}^{n}+\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)D_{+}\tilde{q}_{j}^{n}\\\
&+\frac{1}{2}D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right)\tilde{q}_{j}^{n}\\\
&-\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)\left(\tilde{u}_{j+1/2}^{n}\vee 0\right)\\\
&-\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\wedge
0\right)\left(\tilde{u}_{j-1/2}^{n}\wedge
0\right)+\tilde{P}_{j}^{n}=0.\end{split}$

Multiplying (4.3) by $\tilde{u}_{j+1/2}^{n}$ we find that

(4.5)
$\begin{split}&\tilde{u}_{j+1/2}^{n}D_{+}^{t}\tilde{u}_{j+1/2}^{n}+\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)\left(\tilde{u}_{j+1/2}^{n}\vee 0\right)\tilde{q}_{j}^{n}\\\
&\quad+\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)\left(\tilde{u}_{j+1/2}^{n}\wedge
0\right)\tilde{q}_{j+1}^{n}+D_{+}\tilde{P}_{j}^{n}\tilde{u}_{j+1/2}^{n}=0,\end{split}$

while multiplying (4.4) by $\tilde{q}_{j}^{n}$ gives us

(4.6)
$\begin{split}\tilde{q}_{j}^{n}D_{+}^{t}\tilde{q}_{j}^{n}&+\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\vee
0\right)(D_{-}\tilde{q}_{j}^{n})\tilde{q}_{j}^{n}+\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)(D_{+}\tilde{q}_{j}^{n})\tilde{q}_{j}^{n}\\\
&+\frac{1}{2}D_{-}\left(f^{M}(\tilde{u}_{j+1/2}^{n})\right)\left(\tilde{q}_{j}^{n}\right)^{2}\\\
&-\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)\left(\tilde{u}_{j+1/2}^{n}\vee 0\right)\tilde{q}_{j}^{n}\\\
&-\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\wedge
0\right)\left(\tilde{u}_{j-1/2}^{n}\wedge
0\right)\tilde{q}_{j}^{n}+\tilde{P}_{j}^{n}\tilde{q}_{j}^{n}=0.\end{split}$

Adding (4.5) and (4.6), multiplying the result with ${\Delta x}$, and summing
over $j$ yields

${\Delta
x}\sum_{j\in\mathbb{Z}}\left(\tilde{u}_{j+1/2}^{n}D_{+}^{t}\tilde{u}_{j+1/2}^{n}+\tilde{q}_{j}^{n}D_{+}^{t}\tilde{q}_{j}^{n}\right)+\text{I}+\text{II}+\text{III}=0,$

where

I $\displaystyle={\Delta
x}\sum_{j\in\mathbb{Z}}\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\vee
0\right)(D_{-}\tilde{q}_{j}^{n})\tilde{q}_{j}^{n}$
$\displaystyle\qquad+{\Delta
x}\sum_{j\in\mathbb{Z}}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)(D_{+}\tilde{q}_{j}^{n})\tilde{q}_{j}^{n}+\frac{{\Delta
x}}{2}\sum_{j\in\mathbb{Z}}D_{-}\left(f^{M}\left(u_{j+1/2}\right)\right)\left(\tilde{q}_{j}^{n}\right)^{2},$
II $\displaystyle={\Delta
x}\sum_{j\in\mathbb{Z}}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)\left(\tilde{u}_{j+1/2}^{n}\vee 0\right)\tilde{q}_{j}^{n}$
$\displaystyle\qquad+{\Delta
x}\sum_{j\in\mathbb{Z}}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)\left(\tilde{u}_{j+1/2}^{n}\wedge 0\right)\tilde{q}_{j+1}^{n}$
$\displaystyle\qquad-{\Delta
x}\sum_{j\in\mathbb{Z}}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)\left(\tilde{u}_{j+1/2}^{n}\vee 0\right)\tilde{q}_{j}^{n}$
$\displaystyle\qquad-{\Delta
x}\sum_{j\in\mathbb{Z}}\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\wedge
0\right)\left(\tilde{u}_{j-1/2}^{n}\wedge 0\right)\tilde{q}_{j}^{n}\equiv
0\quad\text{(by shifting indices)},$ III $\displaystyle={\Delta
x}\sum_{j\in\mathbb{Z}}D_{+}\tilde{P}_{j}^{n}\tilde{u}_{j+1/2}^{n}+{\Delta
x}\sum_{j\in\mathbb{Z}}\tilde{P}_{j}^{n}\tilde{q}_{j}^{n}\equiv
0\quad\text{(by summation by parts, cf.~{}\eqref{eq:qjdef}).}$

Let us now deal with term I. The discrete chain rule (2.3) tells us that

$(D_{\pm}\tilde{q}_{j}^{n})\tilde{q}_{j}^{n}=D_{\pm}\left(\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}\right)\mp\frac{{\Delta
x}}{2}(D_{\pm}\tilde{q}_{j}^{n})^{2}.$

Hence

I $\displaystyle={\Delta
x}\sum_{j\in\mathbb{Z}}\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\vee
0\right)\left[D_{-}\left(\frac{(\tilde{q}_{j}^{n})^{2}}{2}\right)+\frac{{\Delta
x}}{2}(D_{-}\tilde{q}_{j}^{n})^{2}\right]$ $\displaystyle\qquad+{\Delta
x}\sum_{j\in\mathbb{Z}}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)\left[D_{+}\left(\frac{(\tilde{q}_{j}^{n})^{2}}{2}\right)-\frac{{\Delta
x}}{2}(D_{+}\tilde{q}_{j}^{n})^{2}\right]$ $\displaystyle\qquad+\frac{{\Delta
x}}{2}\sum_{j\in\mathbb{Z}}D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right)(\tilde{q}_{j}^{n})^{2}=\text{I}_{1}+\text{I}_{2},$

where

$\displaystyle\text{I}_{1}$ $\displaystyle={\Delta
x}\sum_{j\in\mathbb{Z}}\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\vee
0\right)D_{-}\left(\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}\right)$
$\displaystyle\qquad+{\Delta
x}\sum_{j\in\mathbb{Z}}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)D_{+}\left(\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}\right)$
$\displaystyle\qquad+\frac{{\Delta
x}}{2}\sum_{j\in\mathbb{Z}}D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right)(\tilde{q}_{j}^{n})^{2},$
$\displaystyle\text{I}_{2}$ $\displaystyle=\frac{{\Delta
x}^{2}}{2}\sum_{j\in\mathbb{Z}}\Bigl{[}\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\vee
0\right)(D_{-}\tilde{q}_{j}^{n})^{2}-\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)(D_{+}\tilde{q}_{j}^{n})^{2}\Bigr{]}$ $\displaystyle=\frac{{\Delta
x}^{2}}{2}\sum_{j\in\mathbb{Z}}\Bigl{[}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)(D_{+}\tilde{q}_{j}^{n})^{2}-\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)(D_{+}\tilde{q}_{j}^{n})^{2}\Bigr{]}$ $\displaystyle=\frac{{\Delta
x}^{2}}{2}\sum_{j\in\mathbb{Z}}\left|f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right|\left(D_{+}\tilde{q}_{j}^{n}\right)^{2}\geq
0.$

To handle the $\text{I}_{1}$-term, we use the discrete product rule (2.2):

$\displaystyle D_{-}\left[\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}\right]$
$\displaystyle\quad=\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\vee
0\right)D_{-}\left(\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}\right)+D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2},$ $\displaystyle
D_{+}\left[\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\wedge
0\right)\frac{(\tilde{q}_{j}^{n})^{2}}{2}\right]$
$\displaystyle\quad=\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)D_{+}\left(\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}\right)+D_{+}\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\wedge
0\right)\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}$
$\displaystyle\quad=\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)D_{+}\left(\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}\right)+D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}.$

Using this we find that

$\displaystyle\text{I}_{1}$ $\displaystyle={\Delta
x}\sum_{j\in\mathbb{Z}}D_{-}\left[\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}\right]$
$\displaystyle\qquad-{\Delta
x}\sum_{j\in\mathbb{Z}}D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}$
$\displaystyle\qquad+{\Delta
x}\sum_{j\in\mathbb{Z}}D_{+}\left[\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\wedge
0\right)\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}\right]$
$\displaystyle\qquad-{\Delta
x}\sum_{j\in\mathbb{Z}}D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}$
$\displaystyle\qquad+\frac{{\Delta
x}}{2}\sum_{j\in\mathbb{Z}}D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right)\left(\tilde{q}_{j}^{n}\right)^{2}$
$\displaystyle=-{\Delta
x}\sum_{j\in\mathbb{Z}}D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right)\frac{\left(\tilde{q}_{j}^{n}\right)^{2}}{2}$
$\displaystyle\qquad\qquad+\frac{{\Delta
x}}{2}\sum_{j\in\mathbb{Z}}D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right)\left(\tilde{q}_{j}^{n}\right)^{2}=0.$

Summarizing our findings so far:

(4.7) $\begin{split}&{\Delta
x}\sum_{j\in\mathbb{Z}}\left(\tilde{u}_{j+1/2}^{n}D_{+}^{t}\tilde{u}_{j+1/2}^{n}+\tilde{q}_{j}^{n}D_{+}^{t}\tilde{q}_{j}^{n}\right)\\\
&\qquad+\frac{{\Delta
x}^{2}}{2}\sum_{j\in\mathbb{Z}}\left|f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right|\left(D_{+}\tilde{q}_{j}^{n}\right)^{2}=0.\end{split}$

Next, by (2.3),

(4.8) $\begin{split}&{\Delta
x}\sum_{j\in\mathbb{Z}}\left(\tilde{u}_{j+1/2}^{n}D_{+}^{t}\tilde{u}_{j+1/2}^{n}+q_{j}D_{+}^{t}\tilde{q}_{j}^{n}\right)\\\
&\quad=D_{+}^{t}\left[\frac{{\Delta
x}}{2}\sum_{j\in\mathbb{Z}}\left((\tilde{u}_{j+1/2}^{n})^{2}+(\tilde{q}_{j}^{n})^{2}\right)\right]\\\
&\quad\quad\quad-\frac{1}{2}{\Delta t}{\Delta
x}\sum_{j\in\mathbb{Z}}\left(\left(D_{+}^{t}\tilde{u}_{j+1/2}^{n}\right)^{2}+\left(D_{+}^{t}\tilde{q}_{j}^{n}\right)^{2}\right).\end{split}$

Hence, we must now estimate

${\Delta t}{\Delta
x}\sum_{j\in\mathbb{Z}}\left(\left(D_{+}^{t}\tilde{u}_{j+1/2}^{n}\right)^{2}+\left(D_{+}^{t}\tilde{q}_{j}^{n}\right)^{2}\right).$

Using (4.3), (4.4), and the basic inequality
$\left(\sum_{\ell=1}^{l}a_{l}\right)^{2}\leq
2^{l-1}\sum_{\ell=1}^{l}(a_{\ell})^{2}$, which holds for any sequence
$\left\\{a_{\ell}\right\\}_{\ell=1}^{l}$ of positive real numbers, there is a
positive constant $c_{1}$ that does not depend on ${\Delta x}$ such that

$\displaystyle{\Delta t}{\Delta
x}\sum_{j\in\mathbb{Z}}\left(\left(D_{+}^{t}\tilde{u}_{j+1/2}^{n}\right)^{2}+\left(D_{+}^{t}\tilde{q}_{j}^{n}\right)^{2}\right)$
$\displaystyle\qquad\leq c_{1}{\Delta t}{\Delta
x}\sum\limits_{j\in\mathbb{Z}}\Biggl{[}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)^{2}\left(\tilde{q}_{j}^{n}\right)^{2}$
$\displaystyle\quad\qquad\qquad\qquad\qquad+\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)^{2}\left(\tilde{q}_{j+1}^{n}\right)^{2}+\left(D_{+}\tilde{P}_{j}^{n}\right)^{2}$
$\displaystyle\quad\qquad\qquad\qquad\qquad+\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\vee
0\right)^{2}\left(D_{-}\tilde{q}_{j}^{n}\right)^{2}$
$\displaystyle\quad\qquad\qquad\qquad\qquad+\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)^{2}\left(D_{+}\tilde{q}_{j}^{n}\right)^{2}$
$\displaystyle\quad\qquad\qquad\qquad\qquad+\left(D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right)\right)^{2}\left(\tilde{q}_{j}^{n}\right)^{2}$
$\displaystyle\quad\qquad\qquad\qquad\qquad+\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)^{2}\left(\tilde{u}_{j+1/2}^{n}\vee 0\right)^{2}$
$\displaystyle\quad\qquad\qquad\qquad\qquad+\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\wedge
0\right)^{2}\left(\tilde{u}_{j-1/2}^{n}\wedge
0\right)^{2}+\left(\tilde{P}_{j}^{n}\right)^{2}\Biggr{]}$
$\displaystyle\qquad\leq c_{1}{\Delta t}\left(J_{1}+J_{2}+J_{3}\right),$

where

$\displaystyle J_{1}$ $\displaystyle={\Delta
x}\sum\limits_{j\in\mathbb{Z}}\Biggl{[}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)^{2}\left(\tilde{q}_{j}^{n}\right)^{2}+\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)^{2}\left(\tilde{q}_{j+1}^{n}\right)^{2}$
$\displaystyle\qquad\qquad\qquad+\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)^{2}\left(\tilde{u}_{j+1/2}^{n}\vee 0\right)^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\wedge
0\right)^{2}\left(\tilde{u}_{j-1/2}^{n}\wedge 0\right)^{2}\Biggr{]},$
$\displaystyle J_{2}$ $\displaystyle={\Delta
x}\sum\limits_{j\in\mathbb{Z}}\Biggl{[}\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\vee
0\right)^{2}\left(D_{-}\tilde{q}_{j}^{n}\right)^{2}$
$\displaystyle\qquad\qquad\quad+\left(f^{M}\left(u_{j+1/2}\right)\wedge
0\right)^{2}\left(D_{+}\tilde{q}_{j}^{n}\right)^{2}+\left(D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right)\right)^{2}\left(\tilde{q}_{j}^{n}\right)^{2}\Biggr{]},$
$\displaystyle J_{3}$ $\displaystyle={\Delta
x}\sum\limits_{j\in\mathbb{Z}}\Biggl{[}\left(D_{+}\tilde{P}_{j}^{n}\right)^{2}+\left(\tilde{P}_{j}^{n}\right)^{2}\Biggr{]}.$

Since $\left|f^{M}(u)\right|\leq M$, the following bounds hold:

(4.9) $J_{1}\leq 2M^{2}{\Delta
x}\sum\limits_{j\in\mathbb{Z}}\left[\left(\tilde{u}_{j+1/2}^{n}\right)^{2}+(\tilde{q}_{j}^{n})^{2}\right]=2M^{2}\left\|\tilde{u}^{n}\right\|_{h^{1}}$

and

(4.10) $\begin{split}J_{2}&=\frac{{\Delta x}}{{\Delta
x}^{2}}\sum\limits_{j\in\mathbb{Z}}\Biggl{[}\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\vee
0\right)^{2}\left(D_{-}\tilde{q}_{j}^{n}{\Delta x}\right)^{2}\\\
&\qquad\qquad\qquad\qquad+\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\wedge
0\right)^{2}\left(D_{+}\tilde{q}_{j}^{n}{\Delta x}\right)^{2}\\\
&\qquad\qquad\qquad\qquad\qquad+\left(D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right){\Delta
x}\right)^{2}\left(\tilde{q}_{j}^{n}\right)^{2}\Biggr{]}\\\ &\leq
c_{2}M^{2}\frac{{\Delta x}}{{\Delta
x}^{2}}\sum\limits_{j\in\mathbb{Z}}(q_{j})^{2}\leq c_{2}\frac{M^{2}}{{\Delta
x}^{2}}\left\|\tilde{u}^{n}\right\|_{h^{1}},\end{split}$

for some constant $c_{2}>0$ independent of ${\Delta x}$.

To estimate $J_{3}$ we use Lemma 2.2, specifically (2.9), which implies that

(4.11) $\begin{split}J_{3}&\leq c_{3}{\Delta
x}\sum\limits_{j\in\mathbb{Z}}\Biggl{[}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\vee
0\right)^{2}\left(\tilde{u}_{j+1/2}^{n}\vee 0\right)^{2}\\\
&\qquad\qquad\qquad\qquad+\left(f^{M}\left(\tilde{u}_{j-1/2}^{n}\right)\wedge
0\right)^{2}\left(\tilde{u}_{j-1/2}^{n}\wedge 0\right)^{2}\\\
&\qquad\qquad\qquad\qquad\qquad+\left(D_{-}\left(f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right)\right)^{2}\left(\tilde{q}_{j}^{n}\right)^{2}\Biggr{]}\\\
&\leq 2c_{3}M^{2}{\Delta
x}\sum\limits_{i\in\mathbb{Z}}\left(\tilde{u}_{j+1/2}^{n}\right)^{2}+c_{3}M^{2}\frac{{\Delta
x}}{{\Delta
x}^{2}}\sum\limits_{i\in\mathbb{Z}}\left(\tilde{q}_{j}^{n}\right)^{2}\\\ &\leq
c_{3}M^{2}\left(1+\frac{1}{{\Delta
x}^{2}}\right)\left\|\tilde{u}^{n}\right\|_{h^{1}},\end{split}$

for some constant $c_{3}>0$ independent of ${\Delta x}$.

Blending (4.9), (4.10), and (4.11) we derive the bound

(4.12) ${\Delta t}{\Delta
x}\sum_{j\in\mathbb{Z}}\left(\left(D_{+}^{t}\tilde{u}_{j+1/2}^{n}\right)^{2}+\left(D_{+}^{t}\tilde{q}_{j}^{n}\right)^{2}\right)\leq
CM^{2}\left({\Delta t}+\frac{{\Delta t}}{{\Delta
x}^{2}}\right)\left\|\tilde{u}^{n}\right\|_{h^{1}},$

where the constant $C$ is independent of ${\Delta x}$.

Combining (4.7), (4.8), and (4.12), it follows that
$\left\\{\tilde{u}_{j+1/2}^{n}\right\\}$ obeys the following discrete energy
estimate:

$D_{+}^{t}\left\|\tilde{u}^{n}\right\|_{h^{1}}^{2}+\underbrace{\frac{{\Delta
x}^{2}}{2}\sum_{j\in\mathbb{Z}}\left|f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)\right|\left(D_{+}\tilde{q}_{j}^{n}\right)^{2}}_{=:Z^{n}}\leq\underbrace{CM^{2}\left({\Delta
t}+\frac{{\Delta t}}{{\Delta
x}^{2}}\right)}_{=:\omega}\left\|\tilde{u}^{n}\right\|_{h^{1}}^{2}.$

By the discrete Gronwall inequality, cf. Lemma 2.1,

$\left\|\tilde{u}^{N_{0}}\right\|_{h^{1}}^{2}+\sum_{n=0}^{N_{0}-1}e^{CM^{2}\omega(t^{N_{0}-1}-t^{n})}Z^{n}\leq
e^{CM^{2}\omega t^{N_{0}-1}}\left\|u^{0}\right\|_{h^{1}}^{2}.$

Choosing $M=\left\|u_{0}\right\|_{H^{1}(\mathbb{R})}$ and recalling the CFL
type condition (4.1), we deduce

$\displaystyle e^{CM^{2}\omega t^{N}}$
$\displaystyle=e^{C\left\|u_{0}\right\|_{H^{1}(\mathbb{R})}^{2}{\Delta
t}^{2}((1+{\Delta x}^{2})/{\Delta x}^{2})N}$ $\displaystyle\leq(1+{\Delta
x}^{\theta})^{N{\Delta t}}$ $\displaystyle\leq e^{t^{N}{\Delta
x}^{\theta}}\leq 2\quad\text{if}\quad{\Delta x}^{\theta}\leq\frac{\log 2}{T}.$

Therefore, in particular

$\left\|\tilde{u}^{n}\right\|_{h^{1}}\leq\sqrt{2}\left\|u^{0}\right\|_{h^{1}}\quad\text{for
$n=0,\dots,N_{0}$.}$

By the discrete Sobolev inquality (2.1), we find that

$\left\|\tilde{u}^{n}\right\|_{\ell^{\infty}}\leq\frac{1}{\sqrt{2}}\left\|\tilde{u}^{n}\right\|_{h^{1}}\leq\left\|u^{0}\right\|_{h^{1}}\leq
M\quad\text{for $n=0,\dots,N_{0}$.}$

This means that $\tilde{u}^{n}$ will “never notice” $f^{M}$, since

$f^{M}\left(\tilde{u}_{j+1/2}^{n}\right)=\tilde{u}_{j+1/2}^{n}\quad\text{for
$j\in\mathbb{Z}$ and $n=0,\dots,N_{0}$.}$

Therefore,

$\tilde{u}_{j+1/2}^{n}=u_{j+1/2}^{n}\quad\text{and}\quad\tilde{P}_{j}^{n}=P_{j}^{n}\quad\text{for
$j\in\mathbb{Z}$ and $n=0,\dots,N_{0}$.}$

Finally, (4.2) follows by noting that

$e^{CM^{2}\omega(t^{N_{0}-1}-t^{n})}\geq 1\quad\text{for
$n=0,\dots,N_{0}-1$.}$

∎

We conclude this section by stating some immediate consequences of (4.2).

######  Lemma 4.2.

For $n=0,\dots,N$,

$\displaystyle\left\|P^{n}\right\|_{\ell^{\infty}},\left\|P^{n}\right\|_{\ell^{1}}$
$\displaystyle\leq C\left\|u_{0}\right\|^{2}_{H^{1}(\mathbb{R})},$
$\displaystyle\left\|D_{+}P^{n}\right\|_{\ell^{\infty}},\left\|D_{+}P^{n}\right\|_{\ell^{1}}$
$\displaystyle\leq C\left\|u_{0}\right\|^{2}_{H^{1}(\mathbb{R})},$

where $C>0$ is a constant independent of ${\Delta x}$.

###### Proof.

This follows immediately from Lemma 2.2, noting that in this case

$f_{j}=\left(u_{j+1/2}^{n}\vee 0\right)^{2}+\left(u_{j-1/2}^{n}\wedge
0\right)^{2}+\frac{1}{2}\left(D_{-}u_{j+1/2}^{n}\right)^{2},$

and thus

$\left\|f_{j}\right\|_{\ell^{1}}\leq\left\|u^{n}\right\|^{2}_{h^{1}}\leq(1+{\Delta
x})^{2}\left\|u^{0}\right\|_{H^{1}(\mathbb{R})}^{2}.$

∎

##  5\. One-sided sup-norm estimate

######  Lemma 5.1.

Assume that ${\Delta t}$ satisfies the CFL type condition (4.1) and that
${\Delta x}$ is sufficiently small. For $n=0,\dots,N$ and $j\in\mathbb{Z}$, we
then have

(5.1) $q_{j}^{n}\leq\frac{2}{t^{n}}+C\left\|u^{0}\right\|_{h^{1}},$

where $C>0$ is a finite constant.

###### Proof.

We can write the difference equation for $\left\\{q_{j}^{n}\right\\}$, see
(3.7), as

$\begin{split}q^{n+1}_{j}&=q_{j}^{n}\left(1-\lambda a-\lambda
b\right)+q_{j-1}^{n}\lambda a+q_{j+1}^{n}\lambda b-{\Delta
t}\frac{{q_{j}^{n}}^{2}}{2}\\\ &\qquad+{\Delta t}\left(\left(u_{j-1/2}^{n}\vee
0\right)^{2}+\left(u_{j+1/2}^{n}\wedge
0\right)^{2}-P_{j}^{n}\right),\end{split}$

where

$a=\lambda\left(u_{j-1/2}^{n}\vee 0\right),\quad
b=-\lambda\left(u_{j+1/2}^{n}\wedge 0\right),\quad\lambda={\Delta t}/{\Delta
x}.$

Now we have uniform bounds on $\left\|u^{n}\right\|_{\ell^{\infty}}$ and
$\left\|P^{n}\right\|_{\ell^{\infty}}$ and thus

(5.2) $q^{n+1}_{j}\leq q_{j}^{n}\left(1-\lambda a-\lambda
b\right)+q_{j-1}^{n}\lambda a+q_{j+1}^{n}\lambda b-{\Delta
t}\frac{{q_{j}^{n}}^{2}}{2}+{\Delta t}L,$

for some finite constant $L\lesssim\left\|u^{0}\right\|_{h^{1}}^{2}$.

Set $\bar{q}^{n}_{j}=\max\left\\{q_{j}^{n},q_{j-1}^{n},q_{j+1}^{n}\right\\}$.
We claim that

(5.3) $q^{n+1}_{j}\leq\bar{q}^{n}_{j}-{\Delta
t}\frac{\left(\bar{q}^{n}_{j}\right)^{2}}{2}+{\Delta t}L,$

if ${\Delta t}$ is chosen sufficiently small.

First we choose ${\Delta t}$ so small that $\lambda(a+b)<1/2$. Then if
$\bar{q}^{n}_{j}=q_{j}^{n}$ the claim follows immediately from (5.2).

Next, assume $\bar{q}^{n}_{j}=q_{j-1}^{n}$. Then note that

$\displaystyle\left(q_{j}^{n}\right)^{2}$
$\displaystyle=\left(q_{j-1}^{n}\right)^{2}+\left(q_{j}^{n}\right)^{2}-\left(q_{j-1}^{n}\right)^{2}$
$\displaystyle=\left(q_{j-1}^{n}\right)^{2}+\frac{q_{j}^{n}+q_{j-1}^{n}}{2}{\Delta
x}D_{-}q_{j}^{n}$
$\displaystyle=\left(q_{j-1}^{n}\right)^{2}+\frac{u_{j+1/2}^{n}-u^{n}_{j-3/2}}{2}D_{-}q_{j}^{n}.$

Since $D_{-}q_{j}^{n}<0$, we find that

$\left(q_{j}^{n}\right)^{2}\leq\left(q_{j-1}^{n}\right)^{2}-\left\|u^{n}\right\|_{\ell^{\infty}}D_{-}q_{j}^{n}.$

Using this we can rephrase (5.2) as

$\displaystyle q^{n+1}_{j}$ $\displaystyle\leq
q_{j}^{n}\left(1-\lambda\left(a+\left\|u^{n}\right\|_{\ell^{\infty}}+b\right)\right)+q_{j-1}^{n}\lambda\left(a+\left\|u^{n}\right\|_{\ell^{\infty}}\right)+q_{j+1}^{n}\lambda
b$ $\displaystyle\qquad\quad-{\Delta
t}\frac{\left(q_{j-1}^{n}\right)^{2}}{2}+{\Delta t}L$
$\displaystyle\leq\bar{q}^{n}_{j}-{\Delta
t}\frac{\left(\bar{q}^{n}_{j}\right)^{2}}{2}+{\Delta
t}L=:F\left(\bar{q}^{n}_{j}\right),$

if $\lambda\left\|u^{n}\right\|_{\ell^{\infty}}<\frac{1}{2}$. The proof of
(5.3) if $\bar{q}^{n}_{j}=q_{j+1}^{n}$ is similar.

Note that $F^{\prime}(q)=1-{\Delta t}q$, and thus $F$ is increasing for
$q<1/{\Delta t}$. Furthermore, by the CFL type condition (4.1), ${\Delta
t}=\mathcal{O}({\Delta x}^{3})$, and by the bounds on
$\left\|u^{n}\right\|_{\ell^{\infty}}$,

$q_{j}^{n}\leq\left|q_{j}^{n}\right|\leq\mathcal{O}\left(\frac{1}{{\Delta
x}}\right)=\mathcal{O}\left(\frac{1}{{\Delta
t}^{1/3}}\right)\leq\frac{1}{{\Delta t}},$

for sufficiently small ${\Delta t}$. Therefore, setting111This maximum exists
since $q^{n}\in\ell^{2}$. $M^{n}=\max_{j}q_{j}^{n}$, from (5.3) we get

$M^{n+1}\leq F\left(M^{n}\right).$

Now set $Z^{n}=M^{n}-\sqrt{2L}$. Then

$\displaystyle Z^{n+1}$ $\displaystyle\leq
F\left(Z^{n}+\sqrt{2L}\right)-\sqrt{2L}$
$\displaystyle=Z^{n}+\sqrt{2L}-\frac{{\Delta
t}}{2}\left(Z^{n}+\sqrt{2L}\right)^{2}+L{\Delta t}-\sqrt{2L}$
$\displaystyle=Z^{n}\left(1-{\Delta t}\sqrt{2L}\right)-\frac{{\Delta
t}}{2}\left(Z^{n}\right)^{2}$ $\displaystyle\leq Z^{n}-\frac{{\Delta
t}}{2}\left(Z^{n}\right)^{2}.$

Now, clearly if $Z^{n}\leq 0$, then $Z^{n+1}\leq 0$. Hence, if $Z^{0}\leq 0$,
then $Z^{n}\leq 0$ for all $n>0$. If $Z^{n}>0$, by [38, page 271],

$Z^{n}\leq\frac{2}{t^{n}+1/Z^{0}}\leq\frac{2}{t^{n}}.$

This finishes the proof. ∎

##  6\. Higher integrability estimate

We begin this section by deriving a “renormalized form” of the finite
difference scheme for $q_{j}$, so let $f$ be a nonlinear function
(renormalization) of appropriate regularity and growth. Multiplying (3.7) by
$f^{\prime}(q_{j})$ and using the discrete chain rule, which in the present
context reads

$\begin{split}f^{\prime}(q_{j}^{n})D_{\pm}q_{j}^{n}&=D_{\pm}f(q_{j}^{n})\mp\frac{{\Delta
x}}{2}f^{\prime\prime}\left(q^{n}_{j\pm
1/2}\right)\left(D_{\pm}q_{j}^{n}\right)^{2},\\\
f^{\prime}(q_{j}^{n})D_{+}^{t}q_{j}^{n}&=D_{+}^{t}f\left(q_{j}^{n}\right)-\frac{{\Delta
t}}{2}f^{\prime\prime}\left(q^{n+1/2}_{j}\right)\left(D_{+}^{t}q_{j}^{n}\right)^{2},\end{split}$

where $q^{n}_{j\pm 1/2}$ is a number between $q_{j}$ and $q_{j\pm 1}$, and
$q^{n+1/2}_{j}$ is a number between $q_{j}^{n}$ and $q^{n+1}_{j}$. Multiplying
the scheme (3.7) with $f^{\prime}(q_{j}^{n})$ we obtain

(6.1) $\begin{split}D_{+}^{t}f\left(q_{j}^{n}\right)&+\left(u_{j-1/2}^{n}\vee
0\right)D_{-}f(q_{j}^{n})+\left(u_{j+1/2}^{n}\wedge
0\right)D_{+}f(q_{j}^{n})\\\
&+\frac{\left(q_{j}^{n}\right)^{2}}{2}f^{\prime}(q_{j}^{n})+\left[P_{j}^{n}-\left(u_{j+1/2}^{n}\vee
0\right)^{2}-\left(u_{j-1/2}^{n}\wedge
0\right)^{2}\right]f^{\prime}(q_{j}^{n})\\\ &\qquad+I_{{\Delta
x},f^{\prime\prime},j}=\frac{{\Delta
t}}{2}f^{\prime\prime}\left(q^{n+1/2}_{j}\right)\left(D_{+}^{t}q_{j}^{n}\right)^{2},\end{split}$

where

$\displaystyle I_{{\Delta x},f^{\prime\prime},j}$
$\displaystyle:=\frac{{\Delta x}}{2}\Biggl{\\{}\left(u_{j-1/2}^{n}\vee
0\right)f^{\prime\prime}(q^{n}_{j-1/2})(D_{-}q_{j}^{n})^{2}$
$\displaystyle\qquad\qquad\quad-\left(u_{j+1/2}^{n}\wedge
0\right)f^{\prime\prime}(q^{n}_{j+1/2})(D_{+}q_{j}^{n})^{2}\Biggr{\\}}.$

Let us now write (6.1) in divergence-form. To this end, observe that the
discrete product rule (2.2) implies the following relations:

$\displaystyle D_{-}\left[\left(u_{j+1/2}^{n}\vee 0\right)f(q_{j}^{n})\right]$
$\displaystyle=\left(u_{j-1/2}^{n}\vee
0\right)D_{-}f(q_{j}^{n})+D_{-}\left(u_{j+1/2}^{n}\vee 0\right)f(q_{j}^{n}),$
$\displaystyle D_{+}\left[\left(u_{j-1/2}^{n}\wedge
0\right)f(q_{j}^{n})\right]$ $\displaystyle=\left(u_{j+1/2}^{n}\wedge
0\right)D_{+}f(q_{j}^{n})+D_{+}\left(u_{j-1/2}^{n}\wedge 0\right)f(q_{j}^{n})$
$\displaystyle=\left(u_{j+1/2}^{n}\wedge
0\right)D_{+}f(q_{j}^{n})+D_{-}\left(u_{j+1/2}^{n}\wedge
0\right)f(q_{j}^{n}),$

and therefore, using that $q_{j}^{n}=D_{-}u_{j+1/2}^{n}$,

$\displaystyle\left(u_{j-1/2}^{n}\vee
0\right)D_{-}f(q_{j}^{n})+\left(u_{j+1/2}^{n}\wedge 0\right)D_{+}f(q_{j}^{n})$
$\displaystyle\qquad=D_{-}\left[\left(u_{j+1/2}^{n}\vee
0\right)f(q_{j}^{n})\right]+D_{+}\left[\left(u_{j-1/2}^{n}\wedge
0\right)f(q_{j}^{n})\right]-q_{j}^{n}f(q_{j}^{n}).$

Hence, we end up with the following divergence-form variant of the
renormalized difference scheme (6.1):

(6.2) $\begin{split}D_{+}^{t}f(q_{j}^{n})&+D_{-}\left[\left(u_{j+1/2}^{n}\vee
0\right)f(q_{j}^{n})\right]+D_{+}\left[\left(u_{j-1/2}^{n}\wedge
0\right)f(q_{j}^{n})\right]\\\
&+\frac{\left(q_{j}^{n}\right)^{2}}{2}f^{\prime}(q_{j}^{n})-q_{j}^{n}f(q_{j}^{n})\\\
&\qquad+\left[P_{j}^{n}-\left(u_{j+1/2}^{n}\vee
0\right)^{2}-\left(u_{j-1/2}^{n}\wedge
0\right)^{2}\right]f^{\prime}(q_{j}^{n})\\\ &\qquad\qquad+I_{{\Delta
x},f^{\prime\prime},j}=\frac{{\Delta
t}}{2}f^{\prime\prime}\left(q^{n+1/2}_{j}\right)\left(D_{+}^{t}q_{j}^{n}\right)^{2}.\end{split}$

To ensure that the limit of $(\partial_{x}u_{{\Delta x}})^{2}$, cf. (4.2), is
nonsingular (i.e., not a measure), we shall need the following higher
integrability estimate:

######  Lemma 6.1.

Let $q_{{\Delta x}}$ be defined by (3.9), and assume that the CFL type
condition (4.1) holds. Then, for all finite numbers $a,b,\alpha$ with $a<b$
and $\alpha\in(0,\theta)$,

(6.3) $\int_{0}^{T}\int_{a}^{b}\left|q_{{\Delta
x}}\right|^{2+\alpha}\,dx\,dt\leq C,$

for some constant $C=C(a,b,T,\alpha)$ that is independent of ${\Delta x}$.

###### Proof.

Define $\eta_{\varepsilon}(q)=\sqrt{\varepsilon^{2}+q^{2}}-\varepsilon$. Note
that $\eta_{\varepsilon}(q)\approx\left|q\right|$ for small $\varepsilon$ and

$-1<\eta_{\varepsilon}^{\prime}(q)=\frac{q}{\sqrt{\varepsilon^{2}+q^{2}}}<1\quad\text{and}\quad
0<\eta_{\varepsilon}^{\prime\prime}(q)=\frac{\varepsilon^{2}}{(\varepsilon^{2}+q^{2})^{3/2}}\leq\frac{1}{\varepsilon}.$

In (6.2), we then specify

$f(q)=\eta_{\varepsilon}(q)\left(1+q^{2}\right)^{\alpha/2}.$

One can easily check that

$\displaystyle f^{\prime}(q)$
$\displaystyle=\eta_{\varepsilon}^{\prime}(q)\left(1+q^{2}\right)^{\alpha/2}+\alpha\eta_{\varepsilon}(q)q\left(1+q^{2}\right)^{\alpha/2-1}$
$\displaystyle f^{\prime\prime}(q)$
$\displaystyle=\eta_{\varepsilon}^{\prime\prime}(q)\left(1+q^{2}\right)^{\alpha/2}+2\alpha\eta_{\varepsilon}^{\prime}(q)q\left(1+q^{2}\right)^{\alpha/2-1}$
$\displaystyle\qquad\quad+\alpha\eta_{\varepsilon}(q)\left(\left(1+q^{2}\right)^{\alpha/2-1}+2\left(\frac{\alpha}{2}-1\right)\left(1+q^{2}\right)^{\alpha/2-2}\right),$

so that in particular $f^{\prime\prime}(q)\geq 0$ and

(6.4)
$f^{\prime\prime}(q)=\eta_{\varepsilon}^{\prime\prime}(q)\left(1+q^{2}\right)^{\alpha/2}+\text{bounded
terms.}$

Next set

$\displaystyle H(q)$ $\displaystyle:=\frac{q^{2}}{2}f^{\prime}(q)-qf(q)$
$\displaystyle=\frac{q^{2}}{2}\left(\eta_{\varepsilon}^{\prime}(q)\left(1+q^{2}\right)^{\alpha/2}+\alpha\eta_{\varepsilon}(q)\left(1+q^{2}\right)^{\alpha/2-1}q\right)-q\eta_{\varepsilon}(q)\left(1+q^{2}\right)^{\alpha/2}$
$\displaystyle=q\eta_{\varepsilon}(q)\left(1+q^{2}\right)^{\alpha/2}\left[\frac{q\eta_{\varepsilon}^{\prime}(q)}{2\eta_{\varepsilon}(q)}+\frac{\alpha
q^{2}}{2(1+q^{2})}-1\right]$
$\displaystyle=q\eta_{\varepsilon}(q)\left(1+q^{2}\right)^{\alpha/2}\frac{1}{2}\left[\frac{\alpha
q^{2}}{1+q^{2}}+\frac{\varepsilon}{\sqrt{\varepsilon^{2}+q^{2}}}-1\right]=:H_{\varepsilon}(q)h_{\varepsilon}(q),$

with

$H_{\varepsilon}(q)=q\eta_{\varepsilon}(q)\left(1+q^{2}\right)^{\alpha/2},\qquad
h_{\varepsilon}(q)=\frac{1}{2}\left[\frac{\alpha
q^{2}}{1+q^{2}}+\frac{\varepsilon}{\sqrt{\varepsilon^{2}+q^{2}}}-1\right].$

Now note that

$\lim_{q\to-\infty}h_{\varepsilon}(q)\leq\lim_{q\to-\infty}h_{1}(q)=\frac{\alpha-1}{2}<0.$

Hence, for $\alpha,\varepsilon<1$, we can find a constant $K>0$ such that

(6.5) $h_{\varepsilon}(q)<\frac{\alpha-1}{4}\quad\text{for all $q<-K$}.$

Let us continue by defining the sets

$\begin{gathered}\mathcal{N}^{n}=\left\\{j\in\mathbb{Z}\;\bigm{|}\;q_{j}^{n}<-K\right\\},\quad\mathcal{C}^{n}=\left\\{j\in\mathbb{Z}\;\bigm{|}\;-K\leq
q_{j}^{n}\leq 0\right\\},\\\
\text{and}\quad\mathcal{P}^{n}=\left\\{j\in\mathbb{Z}\;\bigm{|}\;q_{j}^{n}>0\right\\},\end{gathered}$

where $K$ is defined in (6.5). Moreover, let $0\leq\chi(x)\leq 1$ be a smooth
cutoff function satisfying

$\chi(x)=\begin{cases}0,&x<a-1,\\\ 1,&x\in[a,b],\\\ 0,&x>b+1.\end{cases}$

We multiply (6.2) by $\chi(x_{j}){\Delta t}{\Delta x}$ and sum over
$(n,j)\in\left\\{0,\dots,N-1\right\\}\times\mathbb{Z}$ to get

$\displaystyle{\Delta t}{\Delta
x}\sum_{n}\sum_{j\in\mathcal{N}^{n}}h_{\varepsilon}$
$\displaystyle\left(q_{j}^{n}\right)H_{\varepsilon}\left(q_{j}^{n}\right)\chi(x_{j})$
(6.6) $\displaystyle\leq-{\Delta t}{\Delta
x}\sum_{n}\sum_{j\in\mathcal{C}^{n}}h_{\varepsilon}\left(q_{j}^{n}\right)H_{\varepsilon}\left(q_{j}^{n}\right)\chi\left(x_{j}\right)$
(6.7) $\displaystyle\quad-{\Delta t}{\Delta
x}\sum_{n}\sum_{j\in\mathcal{P}^{n}}h_{\varepsilon}\left(q_{j}^{n}\right)H_{\varepsilon}\left(q_{j}^{n}\right)\chi\left(x_{j}\right)$
(6.8) $\displaystyle\quad\begin{aligned} -{\Delta t}{\Delta
x}\sum_{j,n}\Bigl{[}&D_{-}\left(\left(u_{j+1/2}^{n}\vee
0\right)f\left(q_{j}^{n}\right)\right)\chi(x_{j})\\\ &\quad
D_{+}\left(\left(u_{j-1/2}^{n}\wedge
0\right)f\left(q_{j}^{n}\right)\right)\chi(x_{j})\Bigr{]}\end{aligned}$ (6.9)
$\displaystyle\quad-{\Delta t}{\Delta
x}\sum_{n,j}A^{n}_{j}\chi\left(x_{j}\right)$ (6.10)
$\displaystyle\quad+{\Delta
x}\sum_{j}\left(f\left(q^{0}_{j}\right)-f\left(q^{N}_{j}\right)\right)\chi(x_{j})$
(6.11) $\displaystyle\quad+{\Delta x}{\Delta t}\sum_{n,j}\frac{{\Delta
t}}{2}f^{\prime\prime}\left(q^{n+1/2}_{j}\right)\chi(x_{j})\left(D_{+}^{t}q_{j}^{n}\right)^{2},$

where

(6.12) $A^{n}_{j}=P_{j}^{n}-\left(u_{j+1/2}^{n}\vee
0\right)^{2}-\left(u_{j-1/2}^{n}\wedge 0\right)^{2}.$

Now for $j\in\mathcal{N}^{n}$ we have that

$\frac{1-\alpha}{4}\left|q_{j}^{n}\right|\eta_{\varepsilon}\left(q_{j}^{n}\right)\left(1+\left(q_{j}^{n}\right)^{2}\right)^{\alpha/2}\leq
h_{\varepsilon}\left(q_{j}^{n}\right)H_{\varepsilon}\left(q_{j}^{n}\right).$

Therefore

$\displaystyle\frac{1-\alpha}{4}$ $\displaystyle{\Delta t}{\Delta
x}\sum_{n}\sum_{j\in\mathcal{N}^{n}}\left|q_{j}^{n}\right|\eta_{\varepsilon}\left(q_{j}^{n}\right)\left(1+\left(q_{j}^{n}\right)^{2}\right)^{\alpha/2}\chi(x_{j})$
$\displaystyle\leq\left|\text{\eqref{eq:Cterm}}\right|+\left|\text{\eqref{eq:Pterm}}\right|+\left|\text{\eqref{eq:diffterm}}\right|+\left|\text{\eqref{eq:Aterm}}\right|+\left|\text{\eqref{eq:fterm}}\right|+\left|\text{\eqref{eq:fppterm}}\right|.$

We shall now find bounds on all the terms on the right hand side; in what
follows, we let $C$ denote a generic constant independent of ${\Delta x}$,
$\varepsilon$, and $\alpha$.

We start with (6.11). By (3.7)

$\displaystyle\left(D_{+}^{t}q_{j}^{n}\right)^{2}$ $\displaystyle\leq
C\Biggl{[}\left(\left(u_{j-1/2}^{n}\vee
0\right)D_{-}q_{j}^{n}\right)^{2}+\left(\left(u_{j+1/2}^{n}\wedge
0\right)D_{+}q_{j}^{n}\right)^{2}$
$\displaystyle\qquad\qquad+\left(P_{j}^{n}\right)^{2}+\left(u_{j+1/2}^{n}\vee
0\right)^{4}+\left(u_{j-1/2}^{n}\wedge
0\right)^{4}+\left(q_{j}^{n}\right)^{4}\Biggr{]}.$

To bound the “integrals” of these terms we must use the CFL type condition
(4.1), which implies that ${\Delta t}=\mathcal{O}({\Delta x}^{2+\theta})$.
First,

$\displaystyle{\Delta x}{\Delta t}\sum_{j}\left(\left(u_{j-1/2}^{n}\vee
0\right)D_{-}q_{j}^{n}\right)^{2}$ $\displaystyle\quad\leq C{\Delta
x}^{1+\theta}\sum_{j}\left({\Delta x}D_{-}q_{j}^{n}\right)^{2}\leq C{\Delta
x}^{1+\theta}\sum_{j}\left(q_{j}^{n}\right)^{2}\leq C{\Delta x}^{\theta}.$

Therefore

$\displaystyle{\Delta t}{\Delta x}\sum_{n,j}{\Delta
t}\Biggl{[}\left(\left(u_{j-1/2}^{n}\vee
0\right)D_{-}q_{j}^{n}\right)^{2}+\left(\left(u_{j+1/2}^{n}\wedge
0\right)D_{+}q_{j}^{n}\right)^{2}\Biggr{]}\chi(x_{j})$
$\displaystyle\qquad\leq C{\Delta x}^{\theta}T\to 0\quad\text{as ${\Delta
x}\to 0$.}$

We also find that

$\displaystyle{\Delta
x}\sum_{j}\Biggl{[}\left(P_{j}^{n}\right)^{2}+\left(u_{j+1/2}^{n}\vee
0\right)^{4}+\left(u_{j-1/2}^{n}\wedge 0\right)^{4}\Biggr{]}$
$\displaystyle\qquad\leq C{\Delta
x}\sum_{j}\Biggl{[}\left|P_{j}^{n}\right|+\left(u_{j+1/2}^{n}\vee
0\right)^{2}+\left(u_{j-1/2}^{n}\wedge 0\right)^{2}\Biggr{]}$
$\displaystyle\qquad\leq C{\Delta
x}\sum_{j}\Biggl{[}\left|P_{j}^{n}\right|+\left(u_{j+1/2}^{n}\right)^{2}\Biggr{]}\leq
C,$

since $u_{{\Delta x}}$ and $P^{n}_{j}$ are uniformly bounded. Thus

${\Delta t}{\Delta x}\sum_{n,j}{\Delta
t}\left[\left(P_{j}^{n}\right)^{2}+\left(u_{j+1/2}^{n}\vee
0\right)^{4}+\left(u_{j-1/2}^{n}\wedge 0\right)^{4}\right]\chi(x_{j})\to 0,$

as ${\Delta x}\to 0$. Additionally,

$\displaystyle{\Delta x}\sum_{j}{\Delta t}\left(q_{j}^{n}\right)^{4}$
$\displaystyle\leq C{\Delta x}\sum_{j}{\Delta x}^{\theta}\left({\Delta
x}q_{j}^{n}\right)^{2}\left(q_{j}^{n}\right)^{2}$ $\displaystyle\leq C{\Delta
x}^{1+\theta}\sum_{j}\left[\left(u_{j+1/2}^{n}\right)^{2}+\left(u_{j-1/2}^{n}\right)^{2}\right]\left(q_{j}^{n}\right)^{2}$
$\displaystyle\leq C{\Delta
x}^{1+\theta}\sum_{j}\left(q_{j}^{n}\right)^{2}\leq C{\Delta x}^{\theta}.$

Therefore

${\Delta x}{\Delta t}\sum_{j}{\Delta
t}\left(q_{j}^{n}\right)^{4}\chi(x_{j})\leq CT{\Delta x}^{\theta}\to
0\quad\text{as ${\Delta x}\to 0$.}$

Now we have established that

(6.13) ${\Delta x}{\Delta t}\sum_{n,j}{\Delta
t}\left(D_{+}^{t}q_{j}^{n}\right)^{2}\chi(x_{j})=\mathcal{O}\left({\Delta
x}^{\theta}\right)\quad\text{as ${\Delta x}\to 0$.}$

Recalling (6.4) this implies that

$\left|\text{\eqref{eq:fppterm}}\right|\leq C{\Delta x}{\Delta
t}\sum_{n,j}\frac{{\Delta
t}}{\varepsilon}\left(1+\left(q^{n+1/2}_{j}\right)^{2}\right)^{\alpha/2}\left(D_{+}^{t}q_{j}^{n}\right)^{2}\chi(x_{j})+\mathcal{O}\left({\Delta
x}^{\theta}\right).$

When we established (6.13) we always had a “${\Delta x}^{\theta}$ to spare”,
which we can use now. With $\beta=\theta-\alpha>0$, we get

$\displaystyle{\Delta t}{\Delta x}\sum_{n,j}$ $\displaystyle\frac{{\Delta
t}}{\varepsilon}\left(1+\left(q^{n+1/2}_{j}\right)^{2}\right)^{\alpha/2}\left(D_{+}^{t}q_{j}^{n}\right)^{2}$
$\displaystyle\leq C{\Delta x}{\Delta t}\sum_{n,j}\frac{{\Delta
x}^{\beta}}{\varepsilon}{\Delta
x}^{\alpha}\left(1+\left(q^{n+1/2}_{j}\right)^{2}\right)^{\alpha/2}{\Delta
x}^{2}\left(D_{+}^{t}q_{j}^{n}\right)^{2}$ $\displaystyle\leq C{\Delta
x}{\Delta t}\sum_{n,j}\frac{{\Delta x}^{\beta}}{\varepsilon}\left({\Delta
x}^{2}+\left({\Delta x}q^{n+1/2}_{j}\right)^{2}\right)^{\alpha/2}{\Delta
x}^{2}\left(D_{+}^{t}q_{j}^{n}\right)^{2}$ $\displaystyle\leq C\frac{{\Delta
x}^{\beta}}{\varepsilon}{\Delta x}{\Delta t}\sum_{n,j}{\Delta
x}^{2}\left(D_{+}^{t}q_{j}^{n}\right)^{2}\leq C\frac{{\Delta
x}^{\beta}}{\varepsilon},$

since

$\left({\Delta x}^{2}+\left({\Delta
x}q^{n+1/2}_{j}\right)^{2}\right)^{\alpha/2}\leq\left({\Delta
x}^{2}+4\left(\left\|u^{n}\right\|_{\ell^{\infty}}\right)^{2}\right)^{\alpha/2}\leq
C.$

Now choosing $\varepsilon={\Delta x}^{\beta}$, we finally conclude that
$\left|\eqref{eq:fppterm}\right|$ is bounded.

Next we turn to (6.6). For $-K\leq q\leq 0$, we have that
$\left|h_{\varepsilon}(q)H_{\varepsilon}(q)\right|\leq C$, where $C$ is
independent of $\varepsilon$. Therefore

$\left|\text{\eqref{eq:Cterm}}\right|\leq CT(b-a+2).$

To estimate $\left|\text{\eqref{eq:Pterm}}\right|$, observe that

$\left|\text{\eqref{eq:Pterm}}\right|\leq C{\Delta t}{\Delta
x}\sum_{n}\sum_{j\in\mathcal{P}^{n}}\left(1+\left|q_{j}^{n}\right|^{2+\alpha}\right)\chi\left(x_{j}\right)$

Now, equipped with (4.2) and (5.1), it is possible to bound the right-hand
side by a “${\Delta x}$ independent” constant exactly as was done in [12].

Regarding (6.8), observe that

$\displaystyle\biggl{|}{\Delta t}$ $\displaystyle{\Delta
x}\sum\limits_{j,n}D_{-}\left[\left(u_{j+1/2}^{n}\vee
0\right)f\left(q_{j}^{n}\right)\right]\chi(j{\Delta x})\biggr{|}$
$\displaystyle=\bigg{|}{\Delta t}{\Delta
x}\sum\limits_{j,n}\left(u_{j+1/2}^{n}\vee 0\right)D_{+}\chi(j{\Delta
x})f\left(q_{j}^{n}\right)\biggr{|}$ $\displaystyle\leq C{\Delta t}{\Delta
x}\sum\limits_{j,n}\left|u_{j+1/2}^{n}\right|\left|D_{+}\chi(j{\Delta
x})\right|\left(1+\left|q_{j}^{n}\right|^{1+\alpha}\right)$ $\displaystyle\leq
CT\Biggl{(}\sup\limits_{n}\left\|u^{n}\right\|_{\ell^{\infty}}\left\|\left\\{D_{+}\chi(x_{j})\right\\}_{j}\right\|_{\ell^{1}}$
$\displaystyle\qquad\qquad\qquad\qquad+\sup\limits_{n}\left\|\left\\{\left|q_{j}^{n}\right|^{1+\alpha}\right\\}_{j}\right\|_{\ell^{\frac{2}{1+\alpha}}}\left\|\left\\{D_{+}\chi(j{\Delta
x})\right\\}_{j}\right\|_{\ell^{\frac{2}{1-\alpha}}}\Biggr{)}$
$\displaystyle=cT\Biggl{(}\sup\limits_{n}\left\|u^{n}\right\|_{\ell^{\infty}}\left\|\left\\{D_{+}\chi(x_{j})\right\\}_{j}\right\|_{\ell^{1}}+\sup\limits_{n}\left\|q^{n}\right\|^{1+\alpha}_{\ell^{2}}\left\|\left\\{D_{+}\chi(j{\Delta
x})\right\\}_{j}\right\|_{\ell^{\frac{2}{1-\alpha}}}\Biggr{)}.$

Therefore, also $\left|\eqref{eq:diffterm}\right|$ is bounded independently of
${\Delta x}$.

Next we focus on (6.9). Remembering that $\left|f^{\prime}(q)\right|\leq
C(1+\left|q\right|^{\alpha})$, we find

$\displaystyle\left|\text{\eqref{eq:Aterm}}\right|$ $\displaystyle\leq{\Delta
t}{\Delta
x}\sum\limits_{n,j}\left(\left\|P^{n}\right\|_{\ell^{\infty}}+2\left\|u^{n}\right\|_{\ell^{\infty}}^{2}\right)\left|f^{\prime}(q_{j}^{n})\right|\chi(x_{j})$
$\displaystyle\leq C{\Delta t}{\Delta
x}\sum\limits_{n,j}\left(\left\|P^{n}\right\|_{\ell^{\infty}}+2\left\|u^{n}\right\|_{\ell^{\infty}}^{2}\right)\left(1+\left|q_{j}^{n}\right|^{\alpha}\right)\chi(x_{j})$
$\displaystyle\leq
CT\left(\sup\limits_{n}\left\|P^{n}\right\|_{\ell^{\infty}}+2\sup\limits_{n}\left\|u^{n}\right\|_{\ell^{\infty}}^{2}\right)$
$\displaystyle\qquad\quad\times\left(\left\|\left\\{\chi(x_{j})\right\\}_{j}\right\|_{\ell^{1}}+\sup\limits_{n}\left\|\left\\{\left|q_{j}^{n}\right|^{\alpha}\right\\}_{j}\right\|_{\ell^{\frac{2}{\alpha}}}\left\|\left\\{\chi(j{\Delta
x})\right\\}_{j}\right\|_{\ell^{\frac{2}{2-\alpha}}}\right)$
$\displaystyle\leq
CT\left(\sup\limits_{n}\left\|P^{n}\right\|_{\ell^{\infty}}+2\sup\limits_{n}\left\|u^{n}\right\|_{\ell^{\infty}}^{2}\right)$
$\displaystyle\qquad\quad\times\left(\left\|\left\\{\chi(j{\Delta
x})\right\\}_{j}\right\|_{\ell^{1}}+\sup\limits_{n}\left\|q^{n}\right\|^{\alpha}_{\ell^{2}}\left\|\left\\{\chi(j{\Delta
x})\right\\}_{j}\right\|_{\ell^{\frac{2}{2-\alpha}}}\right).$

Finally, keeping in mind that $f\geq 0$, we treat (6.10) as follows:

$\displaystyle\left|\text{\eqref{eq:fterm}}\right|$ $\displaystyle\leq{\Delta
x}\sum_{j}\left(f\left(q^{N}_{j}\right)+f\left(q^{0}_{j}\right)\right)\chi(x_{j})$
$\displaystyle\leq C{\Delta
x}\sum_{j}\left(2+\left|q^{N}_{j}\right|^{\alpha}+\left|q^{0}_{j}\right|^{\alpha}\right)\chi(x_{j})$
$\displaystyle\leq\left\|\left\\{\chi(x_{j})\right\\}_{j}\right\|_{\ell^{1}}$
$\displaystyle\quad+\left(\left\|\left\\{\left|q^{N}_{j}\right|^{1+\alpha}\right\\}_{j}\right\|_{\ell^{\frac{2}{1+\alpha}}}+\left\|\left\\{\left|q^{0}_{j}\right|^{1+\alpha}\right\\}_{j}\right\|_{\ell^{\frac{2}{1+\alpha}}}\right)\left\|\left\\{\chi(x_{j})\right\\}_{j}\right\|_{\ell^{\frac{2}{1-\alpha}}}$
$\displaystyle=\left\|\left\\{\chi(x_{j})\right\\}_{j}\right\|_{\ell^{1}}+\left(\left\|q^{N}\right\|_{\ell^{2}}^{1+\alpha}+\left\|q^{0}\right\|_{\ell^{2}}^{1+\alpha}\right)\left\|\left\\{\chi(x_{j})\right\\}_{j}\right\|_{\ell^{\frac{2}{1-\alpha}}}.$

Summarizing, we have established

${\Delta t}{\Delta
x}\sum_{n}\sum_{j}\left|q_{j}^{n}\right|\eta_{\varepsilon}\left(q_{j}^{n}\right)\left(1+\left(q_{j}^{n}\right)^{2}\right)^{\alpha/2}\leq
C.$

The statement of the lemma follows by noting that

$\left|q\right|^{2+\alpha}\leq\left|q\right|\eta_{\varepsilon}(q)\left(1+q^{2}\right)^{\alpha/2}+\left|q\right|^{1+\alpha},$

and using, in combination with (4.2), the bound

${\Delta t}{\Delta
x}\sum_{j,n}\left|q_{j}^{n}\right|^{1+\alpha}\chi(x_{j})\leq
CT\sup_{n}\left\|q^{n}\right\|_{\ell^{2}}^{1+\alpha}\left\|\left\\{\chi(x_{j})\right\\}_{j}\right\|_{\ell^{\frac{2}{1-\alpha}}}.$

∎

##  7\. Basic convergence results

The purpose of this section is to present some straightforward consequences of
the a priori estimates established in the foregoing sections. More precisely,
we prove that the two sequences $\left\\{u_{{\Delta x}}\right\\}_{{\Delta
x}>0}$, cf. (3.10), and $\left\\{P_{{\Delta x}}\right\\}_{{\Delta x}>0}$, cf.
(3.11), have strongly converging subsequences, starting with the former.

######  Lemma 7.1.

There exists a limit function

$u\in L^{\infty}\left(0,T;H^{1}(\mathbb{R})\right)\cap
C([0,T]\times\mathbb{R}),$

such that along a subsequence as ${\Delta x}\to 0$

(7.1) $\displaystyle u_{{\Delta x}}$
$\displaystyle\overset{\star}{\rightharpoonup}u\quad\text{in
$L^{\infty}\left(0,T;H^{1}(\mathbb{R})\right)$,}$ (7.2) $\displaystyle
u_{{\Delta x}}$ $\displaystyle\to u\quad\text{uniformly in $[a,b]\times[0,T]$,
for any $a<b$.}$

Additionally,

(7.3) $t\mapsto\left\|u(t,\cdot)\right\|_{H^{1}(\mathbb{R})}$ is non-
increasing, and (7.4) $\displaystyle\lim_{t\to 0}u(t,x)=u_{0}(x),\qquad
x\in\mathbb{R}.$

###### Proof.

First we note that for $t\in I^{n}$,

$\displaystyle\int_{\mathbb{R}}(u_{{\Delta x}}(t,x))^{2}\,dx$
$\displaystyle=\sum_{j}\\!\int_{x_{j-1/2}}^{x_{j+1/2}}\\!\\!\\!\left(\frac{1}{{\Delta
x}}(x_{j+1/2}-x)u_{j-1/2}(t)\\!+\\!(x-x_{j-1/2})u_{j+1/2}(t)\right)^{2}\,dx$
$\displaystyle\leq\frac{{\Delta
x}}{2}\sum_{j}\left(\left(u_{j-1/2}(t)\right)^{2}+\left(u_{j+1/2}(t)\right)^{2}\right)$
$\displaystyle\leq\frac{1}{{\Delta t}}\left(\left(t-t^{n}\right){\Delta
x}\sum_{j}\left(u^{n+1}_{j+1/2}\right)^{2}+\left(t^{n+1}-t\right){\Delta
x}\sum_{j}\left(u_{j+1/2}^{n}\right)^{2}\right)$

and

$\displaystyle\int_{\mathbb{R}}\left(\partial_{x}u_{{\Delta
x}}(t,x)\right)^{2}\,dx$ $\displaystyle={\Delta
x}\sum_{j}\left(q_{j}(t)\right)^{2}$ $\displaystyle\leq\frac{1}{{\Delta
t}}\left(\left(t-t^{n}\right){\Delta
x}\sum_{j}\left(q^{n+1}_{j}\right)^{2}+\left(t^{n+1}-t\right){\Delta
x}\sum_{j}\left(q_{j}^{n}\right)^{2}\right).$

Hence,

$\left\|u_{{\Delta
x}}(t,\cdot)\right\|_{H^{1}(\mathbb{R})}^{2}\leq\frac{1}{{\Delta
t}}\left(\left(t-t^{n}\right)\left\|u^{n+1}\right\|_{H^{1}(\mathbb{R})}^{2}+\left(t^{n+1}-t\right)\left\|u^{n}\right\|_{H^{1}(\mathbb{R})}^{2}\right).$

Let $s\in I^{m}$ with $m\leq n$. Then, using (4.2),

$\displaystyle\left\|u_{{\Delta x}}(t,\cdot)\right\|_{H^{1}(\mathbb{R})}^{2}$
$\displaystyle\leq\frac{1}{{\Delta
t}}\left(\left(t-t^{n}\right)\left\|u^{n+1}\right\|_{H^{1}(\mathbb{R})}^{2}+\left(t^{n+1}-t\right)\left\|u^{n}\right\|_{H^{1}(\mathbb{R})}^{2}\right)$
$\displaystyle\leq e^{{\Delta t}(n-m){\Delta x}^{\theta}}\left\|u_{{\Delta
x}}(s,\cdot)\right\|_{H^{1}(\mathbb{R})}^{2}$
$\displaystyle\quad+\frac{e^{{\Delta t}(n-m){\Delta x}^{\theta}}}{{\Delta
t}}\biggl{[}\left(t-t^{n}\right)\left(\left\|u^{m+1}\right\|_{H^{1}(\mathbb{R})}^{2}-\left\|u_{{\Delta
x}}(s,\cdot)\right\|_{H^{1}(\mathbb{R})}^{2}\right)$
$\displaystyle\hphantom{\quad+\frac{1}{{\Delta
t}}\biggl{[}}\quad+\left(t^{n+1}-t\right)\left(\left\|u^{m}\right\|_{H^{1}(\mathbb{R})}^{2}-\left\|u_{{\Delta
x}}(s,\cdot)\right\|_{H^{1}(\mathbb{R})}^{2}\right)\biggr{]}$
$\displaystyle\leq e^{{\Delta t}(n-m){\Delta x}^{\theta}}\left\|u_{{\Delta
x}}(s,\cdot)\right\|_{H^{1}(\mathbb{R})}^{2}$ $\displaystyle\qquad+e^{{\Delta
t}(n-m){\Delta
x}^{\theta}}\left|\left\|u^{m+1}\right\|_{H^{1}(\mathbb{R})}^{2}-\left\|u^{m}\right\|_{H^{1}(\mathbb{R})}^{2}\right|$
$\displaystyle\leq e^{{\Delta t}(n-m){\Delta x}^{\theta}}\left\|u_{{\Delta
x}}(s,\cdot)\right\|_{H^{1}(\mathbb{R})}^{2}$ $\displaystyle\qquad+e^{{\Delta
t}(n-m){\Delta x}^{\theta}}\left(e^{{\Delta t}{\Delta
x}^{\theta}}-1\right)\left\|u^{m}\right\|_{H^{1}(\mathbb{R})}^{2}.$

This implies (7.1) and (7.3).

Next we prove that $\left\\{u_{{\Delta x}}\right\\}_{{\Delta x}>0}$ is
uniformly bounded in $W^{1,2+\alpha}((0,T)\times(a,b))$. We can assume that
$a-1=x_{j_{a}}$ and $b+1=x_{j_{b}}$ for some integers $j_{a}$ and $j_{b}$.
Since $q\mapsto\left|q\right|^{2+\alpha}$ is convex,

(7.5) $\displaystyle\int_{0}^{T}\int_{a}^{b}\left|\partial_{x}u_{{\Delta
x}}\right|^{2+\alpha}\,dx$ $\displaystyle\quad\leq\sum_{n}{\Delta
x}\sum_{j}\left(\left(t^{n+1}-t\right)\left|q_{j}^{n}\right|^{2+\alpha}+\left(t-t^{n}\right)\left|q_{j}^{n}\right|^{2+\alpha}\right)\leq
C,$

for some constant $C=C(\alpha,a,b,u_{0})$, where we have also used (6.3).

Now set $\sigma=(x-x_{j-1/2})/{\Delta x}$. Then, for $x\in I_{j}$, we have

$\displaystyle\left|\partial_{t}u_{{\Delta x}}(t,x)\right|$
$\displaystyle=\left|(1-\sigma)D_{+}^{t}u_{j-1/2}^{n}+\sigma
D_{+}^{t}u_{j+1/2}^{n}\right|$
$\displaystyle\leq(1-\sigma)\left|D_{+}^{t}u_{j-1/2}^{n}\right|+\sigma\left|D_{+}^{t}u_{j+1/2}^{n}\right|.$

Furthermore, by the uniform bounds on $D_{+}P_{j}^{n}$ and $u_{{\Delta x}}$,

$\left|D_{+}^{t}u_{j+1/2}^{n}\right|\leq
C(1+\left|q_{j}^{n}\right|+\left|q^{n}_{j-1}\right|).$

Using this,

$\int_{0}^{T}\int_{a}^{b}\left|\partial_{t}u_{{\Delta
x}}\right|^{2+\alpha}\,dx\leq C\left(1+{\Delta t}\sum_{n=0}^{N}{\Delta
x}\sum_{{j_{a}}}^{{j_{b}}}\left|q_{j}^{n}\right|^{2+\alpha}\right)\leq C.$

Now $\left\\{u_{{\Delta x}}\right\\}_{{\Delta x}>0}\subset
W^{1,2+\alpha}\subset\subset C^{0,\ell}$ on $(0,T)\times(a,b)$ with
$\ell=1-2/(2+\alpha)$. Therefore, along a subsequence, $u_{{\Delta x}}\to u$
uniformly in $(0,T)\times(a,b)$ as ${\Delta x}\to 0$.

Let us show that the limit satisfies the initial condition (7.4). Fix
$\bar{x}\in\mathbb{R}$ and let $t\in(0,1)$. We have $\bar{x}\in I_{j}$ for
some $j$ and $u_{{\Delta x}}(x_{j-1/2},0)=u_{0}(x_{j-1/2})$, so that

$\displaystyle\left|u_{{\Delta x}}(0,\bar{x})-u_{0}(\bar{x})\right|$
$\displaystyle\leq\left|u_{{\Delta x}}(0,\bar{x})-u_{{\Delta
x}}(0,x_{j-1/2})\right|+\left|u_{0}(x_{j-1/2})-u_{0}(\bar{x})\right|$
$\displaystyle\leq C\left(\bar{x}-x_{j-1/2}\right)^{\ell}.$

Consequently,

$\displaystyle\left|u(t,\bar{x})-u_{0}(\bar{x})\right|$
$\displaystyle\leq\left|u(t,\bar{x})-u_{{\Delta x}}(t,\bar{x})\right|$
$\displaystyle\qquad+\left|u_{{\Delta x}}(t,\bar{x})-u_{{\Delta
x}}(0,\bar{x})\right|+\left|u_{{\Delta x}}(0,\bar{x})-u_{0}(\bar{x})\right|$
$\displaystyle\leq\left|u(t,\bar{x})-u_{{\Delta
x}}(t,\bar{x})\right|+Ct^{\ell}+C{\Delta x}^{\ell}.$

Now we can let ${\Delta x}\to 0$ and then $t\to 0$ to conclude that
$u(\bar{x},0)=u_{0}(\bar{x})$. This draws to a close the proof of the lemma. ∎

######  Lemma 7.2.

There exists a limit function

$P\in L^{\infty}(0,T;W^{1,\infty}(\mathbb{R}))\cap
L^{\infty}(0,T;W^{1,1}(\mathbb{R})),$

such that along a subsequence as ${\Delta x}\to 0$

(7.6) $\text{$P_{{\Delta x}}\to P$ in
$L^{p}_{\mathrm{loc}}((0,T)\times\mathbb{R})$},\quad 1\leq p<\infty.$

###### Proof.

By the bounds on $P_{j}^{n}$ in Lemma 4.2, we see that $P_{{\Delta x}}$ is
bounded in $L^{\infty}$ uniformly in ${\Delta x}$. Next we show that
$\left\\{\partial_{t}P_{{\Delta x}}\right\\}_{{\Delta x}>0}$ is bounded in
$L^{1}((0,T)\times\mathbb{R})$. For $t\in[t^{n},t^{n+1})$ and $x\in
I_{j+1/2}$,

$\partial_{t}P_{{\Delta
x}}(t,x)=D_{+}^{t}P_{j}^{n}+(x-x_{j})D_{+}^{t}D_{+}P_{j}^{n}=\left(1-\sigma\right)D_{+}^{t}P_{j}^{n}+\sigma
D_{+}^{t}P^{n}_{j+1},$

where $\sigma=(x-x_{j})/{\Delta x}$. Write
$D_{+}^{t}P_{j}^{n}=X^{n}_{j}+Y^{n}_{j}$, where $X^{n}_{j}$ and $Y^{n}_{j}$
solve

$\displaystyle X^{n}_{j}-D_{-}D_{+}X^{n}_{j}$
$\displaystyle=D_{+}^{t}\left(\left(u_{j-1/2}^{n}\wedge
0\right)^{2}+\left(u_{j+1/2}^{n}\vee 0\right)^{2}\right),$ $\displaystyle
Y^{n}_{j}-D_{-}D_{+}Y^{n}_{j}$
$\displaystyle=\frac{1}{2}D_{+}^{t}\left(\left(q_{j}^{n}\right)^{2}\right).$

Then $\left\|X^{n}\right\|_{\ell^{1}}$ is bounded by the $\ell^{1}$ norm of
the corresponding right hand side avbove. By the discrete chain rule

$D_{+}^{t}\left(u_{j-1/2}^{n}\wedge 0\right)^{2}\leq
2\left(u_{j-1/2}^{n}\wedge 0\right)D_{+}^{t}u_{j-1/2}^{n}+{\Delta
t}\left(D_{+}^{t}u_{j-1/2}^{n}\right)^{2}.$

Estimating the first term here

$\displaystyle\left(u_{j-1/2}^{n}\wedge 0\right)D_{+}^{t}u_{j-1/2}^{n}$
$\displaystyle\qquad=-\left(u_{j-1/2}^{n}\wedge
0\right)\left[\left(u_{j-1/2}^{n}\vee
0\right)q^{n}_{j-1}+\left(u_{j-1/2}^{n}\wedge
0\right)q_{j}^{n}+D_{+}P_{j}^{n}\right]$
$\displaystyle\qquad=-\left(u_{j-1/2}^{n}\wedge
0\right)\left[\left(u_{j-1/2}^{n}\wedge
0\right)q_{j}^{n}+D_{+}P_{j}^{n}\right].$

This means that

$\left\|\left(u^{n}\wedge 0\right)D_{+}^{t}u^{n}\right\|_{\ell^{1}}\leq
C\Biggl{[}\left\|u^{n}\right\|_{\ell^{2}}\left\|q^{n}\right\|_{\ell^{2}}+\left\|D_{+}P^{n}\right\|_{\ell^{1}}\Biggr{]}\leq
C.$

Similarly, we have

$\displaystyle\left(D_{+}^{t}u_{j+1/2}^{n}\right)^{2}$
$\displaystyle\\!\leq\\!C\left[\left(u_{j+1/2}^{n}\vee
0\right)^{2}+\left(q_{j}^{n}\right)^{2}+\left(u_{j+1/2}^{n}\wedge
0\right)^{2}+\left(q^{n}_{j+1}\right)^{2}+\left(D_{+}P_{j}^{n}\right)^{2}\right]$
$\displaystyle\leq
C\left[\left(u_{j+1/2}^{n}\right)^{2}+\left(q_{j}^{n}\right)^{2}+\left(q^{n}_{j+1}\right)^{2}+\left|D_{+}P_{j}^{n}\right|\right],$

and thus $\left\|\left(D_{+}^{t}u^{n}\right)^{2}\right\|_{\ell^{1}}\leq C$. We
have shown that

(7.7)
$\left\|X^{n}\right\|_{\ell^{1}}\leq\left\|\left\\{D_{+}^{t}\left[(u^{n}_{j-1/2}\wedge
0)^{2}+(u^{n}_{j+1/2}\vee 0)^{2}\right]\right\\}_{j}\right\|_{\ell^{1}}\leq
C.$

Next, using (6.2) with $f(q)=q^{2}/2$ we have that

(7.8) $\displaystyle\frac{1}{2}D_{+}^{t}\left(q_{j}^{n}\right)^{2}$
$\displaystyle=\underbrace{-D_{-}\left[\left(u_{j+1/2}^{n}\vee
0\right)\left(q_{j}^{n}\right)^{2}\right]+D_{+}\left[\left(u_{j-1/2}^{n}\wedge
0\right)\left(q_{j}^{n}\right)^{2}\right]}_{a^{n}_{j}}$
$\displaystyle\qquad\qquad\quad+\underbrace{A^{n}_{j}q_{j}^{n}-I_{{\Delta
x},f^{\prime\prime},j}+{\Delta
t}\left(D_{+}^{t}q_{j}^{n}\right)^{2}}_{b^{n}_{j}},$

where $A^{n}_{j}$ is defined in (6.12) and

$I_{{\Delta x},f^{\prime\prime},j}={\Delta x}\left\\{\left(u_{j-1/2}^{n}\vee
0\right)\left(D_{-}q_{j}^{n}\right)^{2}-\left(u_{j+1/2}^{n}\wedge
0\right)\left(D_{+}q_{j}^{n}\right)^{2}\right\\}\geq 0.$

We write $Y^{n}_{j}=Y^{a,n}_{j}+Y^{b,n}_{j}$ where

$Y^{a,n}_{j}=\left(I-D_{-}D_{+}\right)^{-1}a^{n}_{j}\quad\text{and}\quad
Y^{b,n}_{j}=\left(I-D_{-}D_{+}\right)^{-1}b^{n}_{j}.$

Now $\left\|Y^{b,n}\right\|_{\ell^{1}}\leq\left\|b^{n}\right\|_{\ell^{1}}$,
and therefore we compute

$\displaystyle\left\|A^{n}q^{n}\right\|_{\ell^{1}}$ $\displaystyle={\Delta
x}\sum_{j}\left|A^{n}_{j}q_{j}^{n}\right|\leq
C\left\|q^{n}\right\|_{\ell^{2}}\left({\Delta
x}\sum_{j}\left[\left(P_{j}^{n}\right)^{2}+\left(u_{j+1/2}^{n}\right)^{2}\right]\right)^{1/2}$
$\displaystyle\leq C\left\|q^{n}\right\|_{\ell^{2}}\left({\Delta
x}\sum_{j}\left[\left|P_{j}^{n}\right|+\left(u_{j+1/2}^{n}\right)^{2}\right]\right)^{1/2}\leq
C.$

By (6.13), the $L^{1}$ norm of ${\Delta t}(D_{+}^{t}q_{j}^{n})^{2}$ is of the
same order as ${\Delta x}^{\theta}$. Then, summing (7.8) over $n$ and $j$, we
arrive at

${\Delta t}{\Delta x}\sum_{n,j}I_{{\Delta x},f^{\prime\prime},j}\leq
CT+{\Delta
x}\sum_{j}\left[\left(q^{N}_{j}\right)^{2}+\left(q^{0}_{j}\right)^{2}\right]+\mathcal{O}\left({\Delta
x}^{\theta}\right)\leq C.$

This means that

(7.9) ${\Delta t}{\Delta x}\sum_{n,j}\left|Y^{b,n}_{j}\right|\leq C.$

Now let

$L_{j}=h\sum_{i}e^{-\kappa\left|i-j\right|}D_{\pm}K_{i}=-h\sum_{i}D_{\mp}\bigl{(}e^{-\kappa\left|i-j\right|}\bigr{)}K_{i},$

for some sequence $\left\\{K_{j}\right\\}_{j}\in\ell^{1}$. Since

$\left|D_{\pm}e^{-\kappa\left|i-j\right|}\right|\leq
Ce^{-\kappa\left|i-j\right|},$

we get

$\left\|L\right\|_{\ell^{1}}={\Delta x}\sum_{j}\left|L_{j}\right|\leq{\Delta
x}hC\sum_{i,j}e^{-\kappa\left|i-j\right|}\left|K_{i}\right|\leq
C\left\|K\right\|_{\ell^{1}}.$

Using this,

$\left\|Y^{a,n}\right\|_{\ell^{1}}\leq C\left\|q^{n}\right\|_{\ell^{2}}^{2}.$

Combining this with (7.9) and (7.7) we see that

${\Delta t}{\Delta x}\sum_{n,j}\left|D_{+}^{t}P_{j}^{n}\right|\leq C,$

and therefore

$\int_{0}^{T}\int_{\mathbb{R}}\left|\partial_{t}P_{{\Delta
x}}\right|\,dx\,dt\leq C.$

Hence $\left\\{P_{{\Delta x}}\right\\}_{{\Delta x}>0}$ is bounded in
$W^{1,1}((0,T)\times\mathbb{R})$. Combining this with the $L^{\infty}$
estimates found in Lemma 4.2 yields the existence of a convergent subsequence
as claimed in (7.6). ∎

##  8\. Strong convergence result

We now show that the sequence $\left\\{q_{{\Delta x}}\right\\}_{{\Delta
x}>0}$, cf. (3.9), has a strongly converging subsequence. This result is a key
point of the convergence analysis.

######  Lemma 8.1.

Fix $1\leq p<3$ and $1\leq r<1+\frac{\theta}{2}$. Then there exist two
functions $q\in L^{p}_{\mathrm{loc}}((0,T)\times\mathbb{R})$,
$\overline{q^{2}}\in L^{r}_{\mathrm{loc}}((0,T)\times\mathbb{R})$ such that
for a subsequence as ${\Delta x}\to 0$,

(8.1) $\displaystyle\text{$q_{{\Delta x}}\overset{\star}{\rightharpoonup}q$ in
$L^{\infty}(0,T;L^{2}(\mathbb{R}))$},$ $\displaystyle\text{$q_{{\Delta
x}}\rightharpoonup q$ in $L^{p}_{\mathrm{loc}}((0,T)\times\mathbb{R})$},$
(8.2) $\displaystyle\text{$q_{{\Delta x}}^{2}\rightharpoonup\overline{q^{2}}$
in $L^{r}_{\mathrm{loc}}((0,T)\times\mathbb{R})$},$

for all $a,b\in\mathbb{R}$, $a<b$. Moreover,

(8.3) $q^{2}(t,x)\leq\overline{q^{2}}(t,x)$ for a.e.
$(t,x)\in(0,T)\times\mathbb{R}$

and

(8.4) $\partial_{x}u=q\quad\text{in the sense of distributions on
$(0,T)\times\mathbb{R}$.}$

Finally, there is a positive constant $C$ such that

(8.5) $q(t,x)\leq\frac{2}{t}+C\left\|u_{0}\right\|_{H^{1}(\mathbb{R})},\qquad
t\in(0,T),\,x\in\mathbb{R}.$

###### Proof.

Claims (8.1), (8.2) are direct consequences of Lemmas 4.1 and 6.1. Claim (8.3)
is true thanks to (8.2) and the convexity of $g=q^{2}$, cf. Lemma 2.3, while
(8.4) is a consequence of the definitions of $q_{{\Delta x}}$ and $u_{{\Delta
x}}$, cf. (3.9) and (3.10).

We conclude by proving (8.5). Fix $t>0$, and let ${\Delta t}$ be so small that
$t\in I^{n}$ with $n>0$. Not that $n\to\infty$ as ${\Delta t}\to 0$, and
$t^{n}$ and $t^{n+1}$ both tend to $t$. From the definition of $q_{{\Delta
x}}$ and (5.1) we have that

(8.6) $\begin{split}q_{{\Delta
x}}(t,x)&=q_{j}^{n}+(t-t^{n})D_{+}^{t}q_{j}^{n}\\\ &=\frac{t^{n+1}-t}{{\Delta
t}}q_{j}^{n}+\frac{t-t^{n}}{{\Delta t}}q^{n+1}_{j}\\\
&\leq\frac{t^{n+1}-t}{{\Delta
t}}\left(\frac{2}{t^{n}}+\hat{C}\right)+\frac{t-t^{n}}{{\Delta
t}}\left(\frac{2}{t^{n}}+\hat{C}\right)\\\ &=\frac{2}{t}+\hat{C}+2f_{{\Delta
t}}(t),\end{split}$

where $\hat{C}:=C\left\|u_{0}\right\|_{H^{1}(\mathbb{R})}$ and for every
$t\in[t^{n},t^{n+1})$, $x\in I_{j}$, with

$f_{{\Delta t}}(t)=\frac{t-t^{n}}{{\Delta
t}}\frac{1}{t^{n+1}}+\frac{t^{n+1}-t}{{\Delta t}}\frac{1}{t^{n}}-\frac{1}{t}.$

Observe that

$f_{{\Delta t}}^{\prime}(t)=\frac{1}{{\Delta
t}}\left(\frac{1}{t^{n+1}}-\frac{1}{t^{n}}\right)+\frac{1}{t^{2}}=-\frac{1}{t^{n}t^{n+1}}+\frac{1}{t^{2}},$

so $f_{{\Delta t}}^{\prime}(t)=0\Longleftrightarrow
t=\sqrt{t^{n}t^{n+1}}\in(t_{n},t_{n+1})$, and in particular

$\displaystyle\sup\limits_{t\in[t^{n},t^{n+1})}f_{{\Delta t}}(t)$
$\displaystyle=f_{{\Delta
t}}(\sqrt{t^{n}t^{n+1}})=\frac{\left(\sqrt{t^{n+1}}-\sqrt{t^{n}}\right)^{2}}{t^{n}t^{n+1}}$
$\displaystyle\leq\left(\frac{{\Delta
t}}{2\sqrt{t^{n}}}\right)^{2}\frac{1}{t^{n}t^{n+1}}\leq\frac{{\Delta
t}^{2}}{8(t^{n})^{2}t^{n+1}}\to 0.$

Therefore (8.5) follows from (8.1) and (8.6). ∎

In view of the weak convergences stated in (8.1), we have that for any
function $f\in C^{1}(\mathbb{R})$ with $f^{\prime}$ bounded

(8.7) $\begin{split}&f(q_{{\Delta
x}})\overset{\star}{\rightharpoonup}\overline{f(q)}\quad\text{in
$L^{\infty}(0,T;L^{2}(\mathbb{R}))$},\\\ &\text{$f(q_{{\Delta
x}})\rightharpoonup\overline{f(q)}$ in
$L^{p}_{\mathrm{loc}}((0,T)\times\mathbb{R})$, $1\leq p<3$},\end{split}$

where the same subsequence of ${\Delta x}\to 0$ applies to any $f$ from the
specified class.

In what follows, we let $\overline{qf(q)}$ and $\overline{f^{\prime}(q)q^{2}}$
denote the weak limits of $q_{{\Delta x}}f(q_{{\Delta x}})$ and
$f^{\prime}(q_{{\Delta x}})q_{{\Delta x}}^{2}$, respectively, in
$L^{r}_{\mathrm{loc}}((0,T)\times\mathbb{R})$, $1\leq r<\frac{3}{2}$.

######  Lemma 8.2.

For any convex function $f\in C^{1}(\mathbb{R})$ with $f^{\prime}$ bounded we
have that

$\begin{split}\iint_{(0,T)\times\mathbb{R}}&\left(\overline{f(q)}\partial_{t}\varphi+u\overline{f(q)}\partial_{x}\varphi\right)\,dx\,dt+\int_{\mathbb{R}}f(q_{0}(x))\varphi(0,x)\,dx\\\
&\qquad\geq\iint_{(0,T)\times\mathbb{R}}\left(\frac{1}{2}\overline{f^{\prime}(q)q^{2}}-\overline{qf(q)}+\left(P-u^{2}\right)\overline{f^{\prime}(q)}\right)\varphi\,dx\,dt,\end{split}$

for any nonnegative $\varphi\in C^{\infty}_{c}([0,T)\times\mathbb{R})$.

###### Proof.

Set

(8.8) $\varphi_{j}(t)=\frac{1}{{\Delta
x}}\int_{I_{j}}\varphi(t,x)\,dx,\quad\varphi_{j}^{n}=\frac{1}{{\Delta
x}{\Delta t}}\iint_{I_{j}^{n}}\varphi(t,x)\,dx\,dt.$

We multiply (6.2) by ${\Delta x}{\Delta t}\varphi_{j}^{n}$, sum over $n,j$,
and take into account the convexity of $f$. After partial summations, the
final result reads

(8.9) $E_{0}+E_{1}+E_{2}+E_{3}+E_{4}+E_{5}\geq 0,$

where

$\displaystyle E_{0}=$ $\displaystyle{\Delta
x}\sum_{j}f(q_{j}^{0})\varphi_{j}^{0},$ $\displaystyle E_{1}=$
$\displaystyle{\Delta t}{\Delta
x}\sum\limits_{n,j}f(q_{j}^{n})D_{t}^{-}\varphi_{j}^{n},$ $\displaystyle
E_{2}=$ $\displaystyle{\Delta t}{\Delta
x}\sum\limits_{n,j}\left[\left(u_{j+1/2}^{n}\vee
0\right)f(q_{j}^{n})D_{+}\varphi_{j}^{n}+\left(u_{j-1/2}^{n}\wedge
0\right)f(q_{j}^{n})D_{-}\varphi_{j}^{n}\right],$ $\displaystyle E_{3}=$
$\displaystyle-{\Delta t}{\Delta
x}\sum\limits_{n,j}\left[\frac{(q_{j}^{n})^{2}}{2}f^{\prime}(q_{j}^{n})-q_{j}^{n}f(q_{j}^{n})\right]\varphi_{j}^{n},$
$\displaystyle E_{4}=$ $\displaystyle-{\Delta t}{\Delta
x}\sum\limits_{n,j}A_{j}^{n}f^{\prime}(q_{j}^{n})\varphi_{j}^{n},$
$\displaystyle E_{5}=$ $\displaystyle{\Delta t}^{2}{\Delta
x}\sum\limits_{n,j}f^{\prime\prime}\left(q^{n+1/2}_{j}\right)\left(D_{+}^{t}q_{j}^{n}\right)^{2}\varphi_{j}^{n}.$

By (3.8),

$\left\|q_{0}-q_{{\Delta x},0}\right\|_{L^{2}(\mathbb{R})}\to 0\quad\text{as
${\Delta x}\to 0$,}$

where $q_{0}:=\partial_{x}u_{0}$ and $q_{{\Delta x},0}:=\partial_{x}u_{{\Delta
x}}|_{t=0}$, so that

${\Delta
x}\sum_{j}f(q_{j}^{0})\varphi_{j}^{0}\to\int_{\mathbb{R}}f(q_{0}(x))\varphi(0,x)\,dx\quad\text{as
${\Delta x}\to 0$}.$

We split $E_{1}$ in three parts:

(8.10) $E_{1}=E_{1,1}+E_{1,2}+E_{1,3},$

where

$\displaystyle E_{1,1}=$
$\displaystyle\iint_{(0,T)\times\mathbb{R}}f(q_{{\Delta
x}})\partial_{t}\varphi\,dx\,dt,$ $\displaystyle E_{1,2}=$
$\displaystyle\sum\limits_{n,j}\iint_{I_{j}^{n}}\left(f(q_{j}^{n})-f(q_{{\Delta
x}})\right)D_{t}^{-}\varphi_{j}^{n}\,dx\,dt\,$ $\displaystyle E_{1,3}=$
$\displaystyle\sum\limits_{n,j}\iint_{I_{j}^{n}}f(q_{{\Delta
x}})\left(D_{t}^{-}\varphi_{j}^{n}-\partial_{t}\varphi\right)\,dt.$

Due to (8.7)

(8.11)
$E_{1,1}\to\iint_{(0,T)\times\mathbb{R}}\overline{f(q)}\partial_{t}\varphi\,dx\,dt.$

Due to the boundedness of $f^{\prime}$,

(8.12) $\left|f(q_{j}^{n})-f(q_{{\Delta x}})\right|\leq
c_{1}(t-t^{n})|D_{+}^{t}q_{j}^{n}|,$

for each $(t,x)\in[t^{n},t^{n+1})\times I_{j}$, where $c_{1}>0$ is a finite
constant. In view of (6.13),

(8.13) $\begin{split}\left|E_{1,2}\right|&\leq c_{1}{\Delta
x}\sum\limits_{n,j}\left|D_{t}^{-}\varphi_{j}^{n}\right||D_{+}^{t}q_{j}^{n}|\int_{t^{n}}^{t^{n+1}}(t-t^{n})\,dt\\\
&\leq c_{1}{\Delta x}{\Delta
t}^{2}\sum\limits_{n,j}\left|D_{t}^{-}\varphi_{j}^{n}\right||D_{+}^{t}q_{j}^{n}|\leq
c_{1}{\Delta
t}\left\|D_{t}^{-}\varphi\right\|_{\ell^{2}}\left\|D_{+}^{t}q\right\|_{\ell^{2}}\to
0,\end{split}$

as ${\Delta x}\to 0$. Finally, since

$\left|D_{t}^{-}\varphi-\partial_{t}\varphi\right|\leq c_{2}{\Delta x},$

for some constant $c_{2}>0$, we have that

(8.14) $\left|E_{1,3}\right|\leq c_{2}{\Delta
x}\iint_{\mathrm{supp}(\varphi)}\left|f(q_{{\Delta x}})\right|\,dx\,dt\to
0\quad\text{as ${\Delta x}\to 0$.}$

Clearly (8.10), (8.11), (8.13), and (8.14) imply

(8.15)
$E_{1}\to\iint_{(0,T)\times\mathbb{R}}\overline{f(q)}\partial_{t}\varphi\,dx\,dt\quad\text{as
${\Delta x}\to 0$.}$

Next, we split $E_{2}$ into four parts:

(8.16) $E_{2}=E_{2,1}+E_{2,2}+E_{2,3}+E_{2.4},$

where

$\displaystyle E_{2,1}=$ $\displaystyle\iint_{(0,T)\times\mathbb{R}}u_{{\Delta
x}}f(q_{{\Delta x}})\partial_{x}\varphi\,dx\,dt,$ $\displaystyle E_{2,2}=$
$\displaystyle\sum\limits_{n,j}\iint_{I_{j}^{n}}\Biggl{[}\left((u_{j+1/2}^{n}\vee
0)-(u_{{\Delta x}}\vee 0)\right)f(q_{j}^{n})D_{+}\varphi_{j}^{n}$
$\displaystyle\qquad\qquad\qquad+\left((u_{j-1/2}^{n}\wedge 0)-(u_{{\Delta
x}}\wedge 0)\right)f(q_{j}^{n})D_{-}\varphi_{j}^{n}\Biggr{]}\,dx\,dt,$
$\displaystyle E_{2,3}=$
$\displaystyle\sum\limits_{n,j}\iint_{I_{j}^{n}}\Biggl{[}(u_{{\Delta x}}\vee
0)\left(f(q_{j}^{n})-f(q_{{\Delta x}})\right)D_{+}\varphi_{j}^{n}$
$\displaystyle\qquad\qquad\qquad+(u_{{\Delta x}}\wedge
0)\left(f(q_{j}^{n})-f(q_{{\Delta
x}})\right)D_{-}\varphi_{j}^{n}\Biggr{]}\,dx\,dt,$ $\displaystyle E_{2,4}=$
$\displaystyle\sum\limits_{n,j}\iint_{I_{j}^{n}}\Biggl{[}\left(u_{{\Delta
x}}\vee 0\right)f(q_{{\Delta
x}})\left(D_{+}\varphi_{j}^{n}-\partial_{x}\varphi\right)$
$\displaystyle\qquad\qquad\qquad+(u_{{\Delta x}}\wedge 0)f(q_{{\Delta
x}})\left(D_{-}\varphi_{j}^{n}-\partial_{x}\varphi\right)\Biggr{]}\,dx\,dt.$

Due to (8.7),

(8.17)
$E_{2,1}\to\iint_{(0,T)\times\mathbb{R}}\overline{qf(q)}\partial_{x}\varphi\,dx\,dt.$

Using the definition of $u_{{\Delta x}}$,

$\begin{split}\left|u_{j-1/2}^{n}-u_{{\Delta x}}\right|&\leq{\Delta
x}\left(\left|q_{j}^{n}\right|+\left|q_{j}^{n+1}\right|\right),\\\
\left|u_{j-1/2}^{n}-u_{{\Delta x}}\right|&\leq{\Delta
x}q_{j}^{n}\left(\left|q_{j}^{n}\right|+\left|q_{j}^{n+1}\right|\right)+\left|u_{j+1/2}^{n}-u_{j-1/2}^{n}\right|,\end{split}$

so, by Lemmas 4.1 and 7.1,

(8.18) $\begin{split}\left|E_{2,2}\right|&\leq{\Delta x}^{2}{\Delta
t}\sum\limits_{n,j}\left(\left|q_{j}^{n}\right|+\left|q_{j}^{n+1}\right|\right)\left|f(q_{j}^{n})\right|\left(\left|D_{+}\varphi_{j}^{n}\right|+\left|D_{-}\varphi_{j}^{n}\right|\right)\\\
&\qquad\qquad\quad+{\Delta x}{\Delta
t}\sum\limits_{n,j}\left|u_{j+1/2}^{n}-u_{j-1/2}^{n}\right|\left|f(q_{j}^{n})\right|\left|D_{+}\varphi_{j}^{n}\right|\\\
&\leq 2{\Delta
x}\left\|D_{+}\varphi\right\|_{\ell^{\infty}}\left\|q\right\|_{\ell^{2}}\left\|f(q)\right\|_{\ell^{2}}\\\
&\qquad\qquad\quad+\left\|f(q)\right\|_{\ell^{2}}\left\|\left\\{\left(u_{j+1/2}^{n}-u_{j-1/2}^{n}\right)D_{+}\varphi_{j}^{n}\right\\}_{n,j}\right\|_{\ell^{2}}\to
0,\end{split}$

as ${\Delta x}\to 0$. Using (8.12) and (6.13), we deduce

(8.19) $\begin{split}\left|E_{2,3}\right|&\leq c_{1}{\Delta
x}\sum\limits_{n,j}\left(\left|D_{-}\varphi_{j}^{n}\right|+\left|D_{+}\varphi_{j}^{n}\right|\right)\\\
&\qquad\qquad\quad\times\left(\left|u_{j+1/2}^{n}\right|+\left|u_{j-1/2}^{n}\right|\right)\int_{t^{n}}^{t^{n+1}}(t-t^{n})\,dt\\\
&=c_{1}{\Delta x}{\Delta
t}^{2}\sum\limits_{n,j}\left(\left|D_{-}\varphi_{j}^{n}\right|+\left|D_{+}\varphi_{j}^{n}\right|\right)\left(\left|u_{j+1/2}^{n}\right|+\left|u_{j-1/2}^{n}\right|\right)\\\
&\leq 4c_{1}{\Delta
t}\left\|D_{+}\varphi\right\|_{\ell^{2}}\left\|u\right\|_{\ell^{2}}\to
0,\end{split}$

as ${\Delta x}\to 0$. Finally, since
$\left|D_{\pm}\varphi-\partial_{t}\varphi\right|\leq c_{3}{\Delta x}$, for
some constant $c_{3}>0$, we obtain

(8.20) $\left|E_{2,4}\right|\leq c_{3}{\Delta
x}\iint_{\mathrm{supp}(\varphi)}\left|q_{{\Delta x}}\right|\left|f(q_{{\Delta
x}})\right|\,dt\,dx\to 0\quad\text{as ${\Delta x}\to 0$.}$

Note that (8.16), (8.17), (8.18), (8.19), and (8.20) imply

(8.21)
$E_{2}\to\iint_{(0,T)\times\mathbb{R}}\overline{qf(q)}\partial_{x}\varphi\,dx\,dt\quad\text{as
${\Delta x}\to 0$.}$

We split $E_{3}$ into three parts:

(8.22) $E_{3}=E_{3,1}+E_{3,2}+E_{3,3},$

where

$\displaystyle E_{3,1}$
$\displaystyle=-\iint_{(0,T)\times\mathbb{R}}\left[\frac{(q_{{\Delta
x}})^{2}}{2}f^{\prime}(q_{{\Delta x}})-q_{{\Delta x}}f(q_{{\Delta
x}})\right]\varphi\,dx\,dt,$ $\displaystyle E_{3,2}$
$\displaystyle=-\sum\limits_{n,j}\iint_{I_{j}^{n}}\Biggl{[}\frac{(q_{j}^{n})^{2}}{2}f^{\prime}(q_{j}^{n})-q_{j}^{n}f(q_{j}^{n})$
$\displaystyle\qquad\qquad\qquad\qquad\quad-\frac{(q_{{\Delta
x}})^{2}}{2}f^{\prime}(q_{{\Delta x}})+q_{{\Delta x}}f(q_{{\Delta
x}})\Biggr{]}\varphi_{j}^{n}\,dx\,dt,$ $\displaystyle E_{3,3}$
$\displaystyle=-\sum\limits_{n,j}\iint_{I_{j}^{n}}\left[\frac{(q_{{\Delta
x}})^{2}}{2}f^{\prime}(q_{{\Delta x}})-q_{{\Delta x}}f(q_{{\Delta
x}})\right]\left(\varphi_{j}^{n}-\varphi\right)\,dx\,dt,$

Due to (8.7),

(8.23)
$E_{3,1}\to-\iint_{(0,T)\times\mathbb{R}}\left(\frac{\overline{q^{2}f^{\prime}(q)}}{2}-\overline{qf(q)}\right)\varphi\,dx\,dt\quad\text{as
${\Delta x}\to 0$.}$

Using the boundedness of $f^{\prime}$, we can estimate as follows:

$\displaystyle\left|\frac{(q_{j}^{n})^{2}}{2}f^{\prime}(q_{j}^{n})-q_{j}^{n}f(q_{j}^{n})-\frac{(q_{{\Delta
x}})^{2}}{2}f^{\prime}(q_{{\Delta x}})+q_{{\Delta x}}f(q_{{\Delta x}})\right|$
$\displaystyle\leq\frac{\left|(q_{j}^{n})^{2}-q_{{\Delta
x}}^{2}\right|}{2}|f^{\prime}(q_{j}^{n})|+\frac{(q_{{\Delta
x}})^{2}}{2}\left|f^{\prime}(q_{j}^{n})-f^{\prime}(q_{{\Delta x}})\right|$
$\displaystyle\leq\frac{\left|q_{j}^{n}-q_{{\Delta
x}}\right|(\left|q_{j}^{n}\right|+\left|q_{{\Delta
x}}\right|)}{2}\left\|f^{\prime}\right\|_{L^{\infty}}+\frac{(q_{{\Delta
x}})^{2}}{2}\left\|f^{\prime\prime}\right\|_{L^{\infty}}\left|q_{j}^{n}-q_{{\Delta
x}}\right|$
$\displaystyle\leq\frac{(t-t^{n})\left|D_{+}^{t}q_{j}^{n}\right|(\left|q_{j}^{n}\right|+\left|q_{{\Delta
x}}\right|)}{2}\left\|f^{\prime}\right\|_{L^{\infty}}+\frac{(q_{{\Delta
x}})^{2}}{2}\left\|f^{\prime\prime}\right\|_{L^{\infty}}(t-t^{n})\left|D_{+}^{t}q_{j}^{n}\right|.$

Hence, taking into account (6.13) and (6.3),

(8.24) $\begin{split}\left|E_{3,2}\right|&\leq{\Delta
x}\sum\limits_{n,j}\int_{I^{n}}\Biggl{[}\frac{(t-t^{n})\left|D_{+}^{t}q_{j}^{n}\right|(\left|q_{j}^{n}\right|+\left|q_{{\Delta
x}}\right|)}{2}\left\|f^{\prime}\right\|_{L^{\infty}}\\\
&\qquad\qquad\qquad\qquad\qquad+\frac{(q_{{\Delta
x}})^{2}}{2}\left\|f^{\prime\prime}\right\|_{L^{\infty}}(t-t^{n})\left|D_{+}^{t}q_{j}^{n}\right|\Biggr{]}\varphi_{j}^{n}\,dt\\\
&\leq{\Delta x}{\Delta
t}^{2}\sum\limits_{n,j}\left[\frac{\left|D_{+}^{t}q_{j}^{n}\right|(\left|q_{j}^{n}\right|+\left|q_{{\Delta
x}}\right|)}{2}\left\|f^{\prime}\right\|_{L^{\infty}}+\frac{(q_{{\Delta
x}})^{2}}{2}\left\|f^{\prime\prime}\right\|_{L^{\infty}}\left|D_{+}^{t}q_{j}^{n}\right|\right]\varphi_{j}^{n}\\\
&\leq{\Delta
t}\Biggl{[}\frac{\left\|D_{+}^{t}q\right\|_{\ell^{2}}(\left\|q^{n}\right\|_{\ell^{2}}+\left\|q_{{\Delta
x}}\right\|_{L^{2}(\mathbb{R})})}{2}\left\|f^{\prime}\right\|_{L^{\infty}}\\\
&\qquad\qquad\qquad\qquad\qquad+\mathcal{O}\left(\frac{1}{{\Delta
x}}\right)\left\|u\right\|_{\ell^{\infty}}\left\|q\right\|_{\ell^{2}}\left\|D_{+}^{t}q^{n}\right\|_{\ell^{2}}\Biggl{]}\left\|\varphi\right\|_{L^{\infty}}\\\
&\leq{\Delta t}\left[\mathcal{O}\left(\sqrt{\frac{{\Delta x}^{\theta}}{{\Delta
t}}}+\sqrt{\frac{{\Delta x}^{\theta}}{{\Delta t}}}\frac{1}{{\Delta
x}}\right)\right]=\mathcal{O}\left({\Delta x}^{\theta}\right)\to
0\quad\text{as ${\Delta x}\to 0$.}\end{split}$

Since $\left|\varphi_{j}^{n}-\varphi\right|={\mathcal{O}}({\Delta x})$,

(8.25) $\left|E_{3,3}\right|\leq{\mathcal{O}}({\Delta
x})\iint_{\mathrm{supp}\,(\varphi)}\left|\frac{(q_{{\Delta
x}})^{2}}{2}f^{\prime}(q_{{\Delta x}})-q_{{\Delta x}}f(q_{{\Delta
x}})\right|\,dx\,dt\to 0\quad\text{as ${\Delta x}\to 0$}.$

We have that (8.22), (8.23), (8.24), and (8.25) imply

(8.26)
$E_{3}\to\iint_{(0,T)\times\mathbb{R}}\overline{qf(q)}\partial_{x}\varphi\,dx\,dt\quad\text{as
${\Delta x}\to 0$.}$

We split the term $E_{4}$ into three parts:

$E_{4}=E_{4,1}+E_{4,2}+E_{4,3},$

where

$\displaystyle E_{4,1}$
$\displaystyle=-\iint_{(0,T)\times\mathbb{R}}A_{{\Delta
x}}f^{\prime}(q_{{\Delta x}})\varphi\,dx\,dt,$ $\displaystyle E_{4,2}$
$\displaystyle=-\sum\limits_{n,j}\iint_{I_{j}^{n}}\left(A_{j}^{n}f^{\prime}(q_{j}^{n})-A_{{\Delta
x}}f^{\prime}(q_{{\Delta x}})\right)\varphi_{j}^{n}\,dx\,dt,$ $\displaystyle
E_{4,3}$ $\displaystyle=-\sum\limits_{n,j}\iint_{I_{j}^{n}}A_{{\Delta
x}}f^{\prime}(q_{{\Delta x}})\left(\varphi_{j}^{n}-\varphi\right)\,dx\,dt,$

where $A_{{\Delta x}}=P_{{\Delta x}}-\left(u_{{\Delta x}}\right)^{2}$. Lemmas
7.1 and 7.2, cf. also (8.7), imply that

$E_{4,1}\to-\iint_{(0,T)\times\mathbb{R}}\left(P-u^{2}\right)\overline{f^{\prime}(q)}\varphi\,dx\,dt\quad\text{as
${\Delta x}\to 0$}.$

Continuing, it is not hard to see that
$\left|E_{4,2}\right|=\mathcal{O}({\Delta x})\to 0$ as ${\Delta x}\to 0$.
Moreover, since $\varphi_{j}^{n}-\varphi=\mathcal{O}({\Delta x})$, $E_{4,3}\to
0$ as ${\Delta x}\to 0$. Summarizing,

(8.27)
$E_{4}\to-\iint_{(0,T)\times\mathbb{R}}\left(P-u^{2}\right)\overline{f^{\prime}(q)}\varphi\,dx\,dt\quad\text{as
${\Delta x}\to 0$.}$

Finally, regarding $E_{5}$, due to (6.13), we conclude that as ${\Delta x}\to
0$

(8.28)
$\left|E_{5}\right|\leq\left\|\varphi\right\|_{L^{\infty}((0,T)\times\mathbb{R})}{\Delta
t}^{2}{\Delta
x}\sum\limits_{n,j}f^{\prime\prime}\left(q^{n+1/2}_{j}\right)\left(D_{+}^{t}q_{j}^{n}\right)^{2}\varphi_{j}^{n}\to
0.$

The lemma now follows from (8.9), (8.15), (8.21), (8.26), (8.27), and (8.28).
∎

We know that $\left\\{(q_{{\Delta x}})^{2}\right\\}_{{\Delta x}>0}$ is bounded
in $L^{\infty}(0,T;L^{1}(\mathbb{R}))\cap
L_{\mathrm{loc}}^{r}((0,T)\times\mathbb{R})$, for any $1\leq r<1+\theta/2$.
Additionally, using (6.2) with $f(q)=\tfrac{q^{2}}{2}$, we can show that the
mapping $t\mapsto\int_{\mathbb{R}}(q_{{\Delta x}})^{2}\varphi\,dx$ is equi-
continuous on $[0,T]$, for every $\varphi\in C^{\infty}_{c}(\mathbb{R})$.
Hence, in view of Lemma 2.4,

(8.29) $\int_{\mathbb{R}}(q_{{\Delta
x}})^{2}\varphi\,dx\to\int_{\mathbb{R}}\overline{q^{2}}\varphi\,dx\quad\text{uniformly
on $[0,T]$,}$

and

(8.30) $t\mapsto\int_{\mathbb{R}}\overline{q^{2}}\varphi\,dx\quad\text{is
continuous on $[0,T]$.}$

The statements (8.29) and (8.30) hold with $q_{{\Delta x}}^{2}$,
$\overline{q^{2}}$ replaced respectively by $f(q_{{\Delta x}})$,
$\overline{f(q_{{\Delta x}})}$, for any convex function $f\in
C^{1}(\mathbb{R})$ with $f^{\prime}$ bounded.

######  Lemma 8.3.

Let $q$ and $\overline{q^{2}}$ be the weak limits identified in Lemma 8.1.
Then

(8.31)
$\begin{split}&\iint_{(0,T)\times\mathbb{R}}\left(q\partial_{t}\varphi+uq\partial_{x}\varphi\right)\,dx\,dt+\int_{\mathbb{R}}q_{0}(x)\varphi(0,x)\,dx\\\
&\quad=\iint_{(0,T)\times\mathbb{R}}\left(-\frac{1}{2}\overline{q^{2}}+\left(P-u^{2}\right)\right)\varphi\,dx\,dt,\quad\forall\varphi\in
C^{\infty}_{c}((0,T)\times\mathbb{R}).\end{split}$

###### Proof.

Starting from (6.2) with $f(q)=q$, we argue as in the proof of Lemma 8.2 to
conclude the validity of (8.31). ∎

The next lemma tells us that the weak limits in Lemma 8.1 satisfy the initial
data in an appropriate sense.

######  Lemma 8.4.

Let $q$ and $\overline{q^{2}}$ be the weak limits identified in Lemma 8.1.
Then

(8.32) $\begin{split}&\lim_{t\to
0}\int_{\mathbb{R}}q^{2}(t,x)\,dx=\int_{\mathbb{R}}(\partial_{x}u_{0})^{2}\,dx,\\\
&\lim_{t\to
0}\int_{\mathbb{R}}\overline{q^{2}}(t,x)\,dx=\int_{\mathbb{R}}(\partial_{x}u_{0})^{2}\,dx.\end{split}$

###### Proof.

The proof is similar to that in [12]. ∎

We can now wrap up the proof of the strong convergence of $\left\\{q_{{\Delta
x}}\right\\}_{{\Delta x}>0}$.

######  Lemma 8.5.

Let $q$ and $\overline{q^{2}}$ be the weak limits identified in Lemma 8.1.
Then

(8.33) $\overline{q^{2}}(t,x)=q^{2}(t,x)$ for a.e.
$(t,x)\in(0,T)\times\mathbb{R}$.

Consequently, as ${\Delta x}\to 0$ (along a subsequence if necessary)

(8.34) $\text{$q_{{\Delta x}}\to q$ in
$L^{2}_{\mathrm{loc}}((0,T)\times\mathbb{R})$ and a.e.~{}in
$(0,T)\times\mathbb{R}$}.$

###### Proof.

By Lemma 8.2,

(8.35)
$\partial_{t}\overline{f(q)}+\partial_{x}\left(u\overline{f(q)}\right)\leq\overline{qf(q)}-\frac{1}{2}\overline{f^{\prime}(q)q^{2}}+\left(u^{2}-P\right)\overline{f^{\prime}(q)},$

in the sense of distributions on $(0,T)\times\mathbb{R}$, for any convex
function $f\in C^{1}(\mathbb{R})$ with $f^{\prime}$ bounded Moreover, by Lemma
8.3,

(8.36) $\partial_{t}q+\partial_{x}(uq)=\frac{1}{2}\overline{q^{2}}+u^{2}-P,$

in the sense of distributions on $(0,T)\times\mathbb{R}$. Equipped with
(8.35), (8.36), (8.32), and (8.5), we can argue exactly as in Xin and Zhang
[39] to arrive at (8.33). In view of Lemma 2.3, claim (8.34) follows
immediately from (8.33) and (7.5). ∎

We now prove that the limit $u$ satisfies (D.3).

######  Lemma 8.6.

For any $\varphi\in C^{\infty}_{c}((0,T)\times\mathbb{R})$,

(8.37)
$\begin{gathered}\int_{0}^{T}\int_{\mathbb{R}}u\varphi_{t}+\left(\frac{u^{2}}{2}+P\right)\varphi_{x}\,dx\,dt=0,\\\
\int_{0}^{T}\int_{\mathbb{R}}P\left(\varphi-\varphi_{xx}\right)\,dx\,dt=\int_{0}^{T}\int_{\mathbb{R}}\left(u^{2}+\frac{1}{2}\left(\partial_{x}u\right)^{2}\right)\varphi\,dx\,dt\end{gathered}$

###### Proof.

It is not difficult to establish the equation for $P$, since we have already
established that $\partial_{x}u_{{\Delta x}}\to\partial_{x}u$ in
$L^{2}_{\mathrm{loc}}((0,T)\times\mathbb{R})$, cf. (8.34). Indeed, we have

$\displaystyle\int_{0}^{T}$ $\displaystyle\int_{\mathbb{R}}P_{{\Delta
x}}\left(\varphi-\varphi_{xx}\right)\,dx\,dt$
$\displaystyle=\sum_{n,j}\iint_{I^{n}_{j-1/2}}P_{{\Delta
x}}\left(\varphi-\varphi_{xx}\right)\,dx\,dt$
$\displaystyle=\sum_{n,j}P_{j}^{n}\iint_{I^{n}_{j-1/2}}\varphi-\varphi_{xx}\,dx\,dt$
$\displaystyle\qquad+\underbrace{\sum_{n,j}\iint_{I^{n}_{j-1/2}}\left(P_{j}^{n}-P_{{\Delta
x}}\right)\left(\varphi-\varphi_{xx}\right)\,dx\,dt}_{w_{1}}$
$\displaystyle={\Delta t}{\Delta
x}\sum_{n,j}P_{j}^{n}\left(\varphi^{n}_{j}-D_{-}D_{+}\varphi^{n}_{j}\right)$
$\displaystyle\qquad+w_{1}+\underbrace{\sum_{n,j}P_{j}^{n}\iint_{I^{n}_{j-1/2}}\left(\left(\varphi-\varphi^{n}_{j}\right)+\left(\varphi_{xx}-D_{-}D_{+}\varphi^{n}_{j}\right)\right)\,dx\,dt}_{w_{2}}.$

Since $\left|P_{{\Delta x}}(t,x)-P_{j}^{n}\right|\leq C{\Delta x}$ for
$(t,x)\in I^{n}_{j-1/2}$, $w_{1}\to 0$ as ${\Delta x}\to 0$. Similarly, since
$\varphi-\varphi_{xx}$ is close to $\varphi^{n}_{j}-D_{-}D_{+}\varphi^{n}_{j}$
in $I^{n}_{j-1/2}$ and $P_{j}^{n}$ is bounded, we conclude that $w_{2}\to 0$
as ${\Delta x}\to 0$. Set

$\displaystyle f^{n}_{j}$ $\displaystyle=\left(u_{j-1/2}^{n}\wedge
0\right)^{2}+\left(u_{j+1/2}^{n}\vee
0\right)^{2}+\frac{1}{2}\left(q_{j}^{n}\right)^{2},$ $\displaystyle f_{{\Delta
x}}$ $\displaystyle=\left(u_{{\Delta
x}}\right)^{2}+\frac{1}{2}\left(\partial_{x}u_{{\Delta x}}\right)^{2}.$

Using the scheme for $P_{j}^{n}$, cf. (3.3),

$\displaystyle{\Delta t}{\Delta
x}\sum_{n,j}P_{j}^{n}\left(\varphi^{n}_{j}-D_{-}D_{+}\varphi^{n}_{j}\right)={\Delta
t}{\Delta x}\sum_{n,j}f^{n}_{j}\varphi^{n}_{j}$
$\displaystyle\qquad=\int_{0}^{T}\int_{\mathbb{R}}f_{{\Delta
x}}\varphi\,dx\,dt$
$\displaystyle\qquad\qquad+\underbrace{\sum_{n,j}\iint_{I^{n}_{j-1/2}}\left(f^{n}_{j}-f_{{\Delta
x}}\right)\varphi\,dx\,dt}_{w_{3}}+\sum_{n,j}f^{n}_{j}\underbrace{\iint_{I^{n}_{j-1/2}}\left(\varphi-\varphi^{n}_{j}\right)\,dx\,dt}_{=0}.$

By the definition of $q_{{\Delta x}}$ we have that $q_{{\Delta
x}}=q_{j}^{n}+(t-t^{n})D_{+}^{t}q_{j}^{n}$ for $t\in I^{n}$. Hence

$(q_{{\Delta
x}})^{2}-\left(q_{j}^{n}\right)^{2}=2(t-t^{n})q_{j}^{n}D_{+}^{t}q_{j}^{n}+\left(t-t^{n}\right)^{2}\left(D_{+}^{t}q_{j}^{n}\right)^{2}\quad\text{in
$I^{n}_{j-1/2}$.}$

so that, assuming $\mathrm{supp}(\varphi)\subset[x_{j_{a}},x_{j_{b}}]$ for
some intergers $j_{a}$ and $j_{b}$,

$\displaystyle\sum_{n,j}\iint_{I^{n}_{j-1/2}}\left|\left(q_{j}^{n}\right)^{2}-\left(q_{{\Delta
x}}\right)^{2}\right|\left|\varphi\right|\,dx\,dt$ $\displaystyle\quad\leq
C{\Delta
x}\sum_{n}\sum_{j_{a}}^{j_{b}}\int_{t^{n}}^{t^{n+1}}\left(2(t-t^{n})q_{j}^{n}D_{+}^{t}q_{j}^{n}+\left(t-t^{n}\right)^{2}\left(D_{+}^{t}q_{j}^{n}\right)^{2}\right)\,dt$
$\displaystyle\quad=C{\Delta t}{\Delta
x}\sum_{n}\sum_{j=j_{a}}^{j_{b}}\left({\Delta
t}q_{j}^{n}D_{+}^{t}q_{j}^{n}+\left({\Delta
t}D_{+}^{t}q_{j}^{n}\right)^{2}\right)$
$\displaystyle\quad=\mathcal{O}\left(\sqrt{{\Delta t}{\Delta
x}^{\theta}}+{\Delta t}{\Delta x}^{\theta}\right).$

By the Hölder continuity of $u_{{\Delta x}}$; recall that $u_{{\Delta x}}\in
C^{0,\ell}$ with $\ell=1-2/(2+\alpha)$, we find

$\left|\left(u_{j-1/2}^{n}\wedge 0\right)^{2}+\left(u_{j+1/2}^{n}\vee
0\right)^{2}-\left(u_{{\Delta x}}\right)^{2}\right|=\mathcal{O}({\Delta
x}^{\ell}+{\Delta t}^{\ell}),$

and therefore $w_{3}\to 0$ as ${\Delta x}\to 0$. Hence, using (7.2), (8.4),
and (8.34),

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}P\left(\varphi-\varphi_{xx}\right)\,dx\,dt$
$\displaystyle=\lim_{{\Delta x}\to 0}\int_{0}^{T}\int_{\mathbb{R}}P_{{\Delta
x}}\left(\varphi-\varphi_{xx}\right)\,dx\,dt$ $\displaystyle=\lim_{{\Delta
x}\downarrow 0}\int_{0}^{T}\int_{\mathbb{R}}\left(\left(u_{{\Delta
x}}\right)^{2}+\frac{1}{2}\left(\partial_{x}u_{{\Delta
x}}\right)^{2}\right)\varphi\,dx\,dt$
$\displaystyle=\int_{0}^{T}\int_{\mathbb{R}}\left(u^{2}+\frac{1}{2}\left(\partial_{x}u\right)^{2}\right)\varphi\,dx\,dt.$

This means that the second equation in (8.37) holds.

To establish the first equality in (8.37), we derive a divergence-form version
of the scheme (3.1). To this end, introduce the functions
$f_{\vee}(u)=\frac{1}{2}(u\vee 0)^{2}$ and $f_{\wedge}(u)=\frac{1}{2}(u\wedge
0)^{2}$. Observe that $f_{\vee}$ and $f_{\wedge}$ are piecewise $C^{2}$, and
the absolute value of the second derivatives are bounded by $1$. By the
discrete chain rule,

$\left(u_{j+1/2}^{n}\vee
0\right)D_{-}u_{j+1/2}^{n}=D_{-}f_{\vee}(u_{j+1/2}^{n})+\mathcal{O}\left({\Delta
x}\left(D_{-}u_{j+1/2}^{n}\right)^{2}\right)$

and

$\left(u_{j+1/2}^{n}\wedge
0\right)D_{+}u_{j+1/2}^{n}=D_{+}f_{\wedge}(u_{j+1/2}^{n})+\mathcal{O}\left({\Delta
x}\left(D_{+}u_{j+1/2}^{n}\right)^{2}\right).$

Consequently, we can replace (3.1) by

(8.38)
$\begin{split}D_{+}^{t}u_{j+1/2}^{n}&+D_{-}f_{\vee}(u_{j+1/2}^{n})+D_{+}f_{\wedge}(u_{j+1/2}^{n})+D_{+}P_{j}^{n}\\\
&=\mathcal{O}\left({\Delta
x}\left\\{\left(D_{-}u_{j+1/2}^{n}\right)^{2}+\left(D_{+}u_{j+1/2}^{n}\right)^{2}\right\\}\right).\end{split}$

Observe that

(8.39) $D=\frac{D_{-}+D_{+}}{2},\qquad{\Delta x}D_{-}D_{+}=D_{+}-D_{-},\qquad
f_{\vee}+f_{\wedge}=\frac{u^{2}}{2}.$

Using these identities, we can restate (8.38) as

(8.40)
$\begin{split}D_{+}^{t}u_{j+1/2}^{n}&+D_{-}\left[\frac{\left(u_{j+1/2}^{n}\right)^{2}}{4}+\frac{1}{2}\left(f_{\vee}(u_{j+1/2}^{n})-f_{\wedge}(u_{j+1/2}^{n})\right)\right]\\\
&+D_{+}\left[\frac{\left(u_{j+1/2}^{n}\right)^{2}}{4}+\frac{1}{2}\left(f_{\wedge}(u_{j+1/2}^{n})-f_{\vee}(u_{j+1/2}^{n})\right)\right]+D_{+}P_{j}^{n}\\\
&\qquad\qquad={\mathcal{O}}\left({\Delta
x}\left\\{\left(D_{-}u_{j+1/2}^{n}\right)^{2}+\left(D_{+}u_{j+1/2}^{n}\right)^{2}\right\\}\right).\end{split}$

Using, cf. (8.39),

$\displaystyle
D_{-}\left(f_{\vee}(u_{j+1/2}^{n})-f_{\wedge}(u_{j+1/2}^{n})\right)+D_{+}\left(f_{\wedge}(u_{j+1/2}^{n})-f_{\vee}(u_{j+1/2}^{n})\right)$
$\displaystyle\qquad={\Delta x}D_{-}D_{+}f_{\wedge}(u_{j+1/2}^{n})-{\Delta
x}D_{-}D_{+}f_{\vee}(u_{j+1/2}^{n}),$

equation (8.40) becomes

(8.41)
$\begin{split}D_{+}^{t}u_{j+1/2}^{n}&+D\left(\frac{\left(u_{j+1/2}^{n}\right)^{2}}{2}\right)+D_{+}P_{j}^{n}\\\
&=\mathcal{O}\left({\Delta
x}\left\\{\left(D_{-}u_{j+1/2}^{n}\right)^{2}+\left(D_{+}u_{j+1/2}^{n}\right)^{2}\right\\}\right)\\\
&\qquad+{\Delta
x}\left\\{D_{-}D_{+}f_{\vee}(u_{j+1/2}^{n})-D_{-}D_{+}f_{\wedge}(u_{j+1/2}^{n})\right\\}.\end{split}$

Now fix $\varphi\in C^{2}_{c}((0,T)\times\mathbb{R})$ and define
$\varphi^{n}_{j}$ as before, cf. (8.8). Multiplying (8.41) by
$\varphi^{n}_{j}{\Delta t}{\Delta x}$ and performing partial summations gives

$\displaystyle\underbrace{{\Delta t}{\Delta
x}\sum_{n,j}u_{j+1/2}^{n}D_{+}^{t}\varphi^{n}_{j}}_{E_{1}}\\!+\\!\underbrace{{\Delta
t}{\Delta
x}\sum_{n,j}\frac{\left(u_{j+1/2}^{n}\right)^{2}}{2}D\varphi^{n}_{j}}_{E_{2}}\\!+\\!\underbrace{{\Delta
t}{\Delta
x}\sum_{n,j}P_{j}^{n}D_{-}\varphi^{n}_{j}}_{E_{3}}=\mathcal{O}({\Delta x}),$

by using (4.2). We have $\left|u_{{\Delta x}}-u_{j-1/2}^{n}\right|\leq
C\left({\Delta x}\left|q_{j}^{n}\right|+{\Delta
t}\left|D_{+}^{t}u_{j+1/2}^{n}\right|\right)$. Using this and (7.2), we
compute as follows:

$\displaystyle E_{1}$ $\displaystyle=\int_{0}^{T}\int_{\mathbb{R}}u_{{\Delta
x}}\varphi_{t}\,dx\,dt+\sum_{n,j}\iint_{I^{n}_{j}}\left(u_{j+1/2}^{n}-u_{{\Delta
x}}\right)\varphi_{t}\,dx\,dt$
$\displaystyle\qquad\qquad+\sum_{n,j}u_{j+1/2}^{n}\iint_{I^{n}_{j}}\left(D_{+}^{t}\varphi^{n}_{j}-\varphi_{t}\right)\,dx\,dt$
$\displaystyle=\int_{0}^{T}\int_{\mathbb{R}}u_{{\Delta
x}}\varphi_{t}\,dx\,dt+\mathcal{O}({\Delta
x})\to\int_{0}^{T}\int_{\mathbb{R}}u\varphi_{t}\,dx\,dt\quad\text{as ${\Delta
x}\to 0$.}$

In the same way, equipped with (7.2) and (7.6), we can show that

$E_{2}\to\int_{0}^{T}\int_{\mathbb{R}}\frac{u^{2}}{2}\varphi_{x}\,dx\,dt,\quad
E_{3}\to\int_{0}^{T}\int_{\mathbb{R}}P\varphi_{x}\,dx\,dt\quad\text{as
${\Delta x}\to 0$.}$

thus proving the first equality in (8.37). This concludes the proof of the
lemma. ∎

##  9\. Numerical examples

We have tried the difference method presented here on several examples, and in
doing this found that the first order method analyzed in this paper exhibits
very slow convergence, and thus requires a very small mesh size ${\Delta x}$
to compute reasonable solutions. This is not surprising and appears to be the
case with other schemes in the literature as well. Therefore we have
implemented a second order extension of the method. This second order
extension is based on the conservative version of the scheme

(9.1) $D_{+}^{t}u_{j+1/2}^{n}+D_{-}\left[\left(u_{j+1/2}^{n}\vee
0\right)u_{j+1/2}^{n}+\left(u_{j+1/2}^{n}\wedge
0\right)u^{n}_{j+3/2}+P^{n}_{j+1}\right]\\\ =D_{-}\left(u_{j+1/2}^{n}\vee
0\right)u_{j-1/2}^{n}+D_{-}\left(u_{j+1/2}^{n}\wedge 0\right)u_{j+1/2}^{n},$

which can be viewed as a balance equation with a flux across $x=x_{j+1/2}$
given by

$F_{j+1/2}^{n}=\left(u_{j+1/2}^{n}\vee
0\right)u_{j+1/2}^{n}+\left(u_{j+1/2}^{n}\wedge
0\right)u^{n}_{j+3/2}+P^{n}_{j+1}.$

Taking this viewpoint, we _define_ the second order finite volume scheme by

(9.2)
$D_{+}^{t}u_{j}^{n}+D_{-}F_{j+1/2}^{n+1/2}=u_{j}^{n}D_{-}\left(u^{n+1/2}_{j+1/2}\right).$

Here $u^{n+1/2}_{j+1/2}$ is a first order approximation of the value at the
point $x=x_{j+1/2}$, $t=t^{n}+{\Delta t}/2$. This approximation is found by
setting

$u_{j+1/2}^{n}=\frac{1}{2}\left(u^{n}_{j}+u^{n}_{j+1}\right),$

and then using the scheme (3.1) for half a time step (i.e., ${\Delta t}/2$).
This scheme is a formally second order accurate finite volume approximation,
and this simple adaptation produces significantly more accurate
approximations.

In Figure 1 we show the approximations calculated by the first order scheme
(3.1) and the second order scheme (9.2) for the single peakon example. In this
case the exact solution reads

$u(x,t)=e^{-\left|x-t\right|}.$

Figure 1 shows the solutions calculated using $2^{9}$ equally spaced grid
points in the interval $[-10,30]$ for $t=20$. We see that the second order
method is much more accurate than the first order method.

Figure 1. Approximations using ${\Delta x}=40/2^{9}$, at $t=20$ to the single
peakon.

In passing, we note that we have not used the strict CFL-condition (4.1), but
the more natural condition

${\Delta t}\leq\max_{j}\left\\{u^{n}_{j}\right\\}{\Delta x}.$

This holds for the second order scheme as well.

In order to investigate the convergence properties of the two methods, we
computed errors in $L^{1}$ for the two schemes. Table 1 shows the computed
$L^{1}$ errors in the case of a single peakon under mesh refinement. In this
context, the $L^{1}$ error is defined as

$\text{$L^{1}$ error}={\Delta x}\sum_{i}\left|u_{{\Delta
x}}(x_{i},t)-u(x_{i},t)\right|,$

where $u$ is the exact solution. We used $t=20$ and ${\Delta x}=40/2^{k}$ for
$k=5,\ldots,12$. As expected, and as reported in [12], the first order method
converges very slowly.

$k$ | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13
---|---|---|---|---|---|---|---|---|---
$1^{\mathrm{st}}$ | 2.92 | 3.23 | 3.41 | 3.53 | 3.57 | 3.51 | 3.32 | 3.01 | 2.64
$2^{\mathrm{nd}}$ | 5.36 | 5.17 | 3.29 | 1.27 | 0.60 | 0.36 | 0.21 | 0.13 | 0.09
Table 1. $L^{1}$ errors for the single peakon case, at $t=20$, for
$x\in[-10,30]$, ${\Delta x}=40/2^{k}$, $k=5,\ldots,13$

One other notable feature of Table 1 is that the second order method seems to
converge at a rate slightly less than $1$.

The two-peakon solution is considerably more complicated than the single
peakon, and this is also a much harder challenge computationally, see e.g.,
[1] and [33]. We use the two-peakon solution given by

(9.3)
$u(x,t)=m_{1}(t)e^{-\left|x-x_{1}(t)\right|}+m_{2}(t)e^{-\left|x-x_{2}(t)\right|},$

with

$\begin{gathered}x_{1}(t)=\log\left(\frac{18e^{t-10}}{e^{(t-10)/2}+6}\right),\quad
x_{2}(t)=\log\left(40e^{t-10}+60e^{(t-10)/2}\right)\\\
m_{1}(t)=\frac{e^{(t-10)/2}+6}{2e^{(t-10)/2}+3},\quad
m_{2}(t)=\frac{e^{(t-10)/2}+\frac{2}{3}}{e^{(t-10)/2}+3}.\end{gathered}$

These formulas were taken from [35]. Figure 2 shows a contour plot of the
approximate solutions found by using the first and second order methods, and
${\Delta x}=40/2^{10}$ for $x\in[-15,25]$ and $t\in[0,25]$.

|
---|---
Figure 2. Approximations to (9.3) using ${\Delta x}=40/2^{10}$. Left: first
order method(3.1). Right : second order method (9.2).

We see that the interaction between the two peakons is poorly represented by
the first order method. Both the location as well as the magnitude of the
peaks are far from the correct value. This is also illustrated Figure 3 where
we show the approximations using ${\Delta x}=40/2^{8}$ at $t=25$.

Figure 3. The approximations to (9.3) at $t=25$ and ${\Delta x}=40/2^{8}$.

We have also calculated errors for the two-peakon case. Indeed for ${\Delta
x}\geq 40/2^{12}$, the first order method did not seem to converge, and in
order to give meaningful answers, this method demands very fine
discretizations. These results are reported in Table 2. For ${\Delta
x}>40/2^{8}$ none of the methods gave satisfactory results.

$k$ | 8 | 9 | 10 | 11 | 12 | 13
---|---|---|---|---|---|---
$1^{\text{st}}$ | 4.56 | 3.64 | 3.97 | 4.18 | 4.05 | 3.70
$2^{\text{nd}}$ | 1.88 | 1.04 | 0.63 | 0.38 | 0.22 | 0.16
Table 2. $L^{1}$ errors for the approximation to (9.2), $t=25$,
$x\in[-15,25]$, ${\Delta x}=40/2^{k}$, $k=8,\ldots,13$.

In our final example we choose initial data corresponding to a peakon-
antipeakon collision:

(9.4)
$u_{0}(x)=-\tanh(6)\left(e^{-\left|x+y(6)\right|}-e^{-\left|x-y(6)\right|}\right),$

where $y(t)=\log(\cosh(t))$. In this case we have a “peakon anti-peakon
collision” at $t=6$. In Figure 4 we exhibit the approximations generated by
the first order (left) and the second order method for $t\in[0,10]$ and
${\Delta x}=24/2^{12}$. It is clear that the first order scheme generates the
dissipative solution, and for $t$ larger than the collision time, the first
order approximation vanishes. Regarding the second order approximation, it
seems to continue as a peakon moving to the right, and an anti-peakon moving
to the left. The magnitudes and speeds of these features are however far from
the conservative solution, and we have indicated the conservative solution in
the right hand figure.

|
---|---
Figure 4. The numerical solutions to the initial value problem (9.4). Left:
first order method, right: second order version.

## References

  * [1] R. Artebrant and H. J. Schroll.  Numerical simulation of Camassa-Holm peakons by adaptive upwinding.  Appl. Numer. Math., 56(5):695–711, 2006.
  * [2] R. Beals, D. H. Sattinger, and J. Szmigielski.  Multipeakons and the classical moment problem.  Adv. Math., 154(2):229–257, 2000.
  * [3] A. Bressan and A. Constantin.  Global conservative solutions of the Camassa-Holm equation.  Arch. Ration. Mech. Anal., 183(2):215–239, 2007.
  * [4] A. Bressan and A. Constantin.  Global dissipative solutions of the Camassa-Holm equation.  Anal. Appl. (Singap.), 5(1):1–27, 2007.
  * [5] A. Bressan and M. Fonte.  An optimal transportation metric for solutions of the Camassa-Holm equation.  Methods Appl. Anal., 12(2):191–219, 2005.
  * [6] R. Camassa.  Characteristics and the initial value problem of a completely integrable shallow water equation.  Discrete Contin. Dyn. Syst. Ser. B, 3(1):115–139, 2003.
  * [7] R. Camassa and D. D. Holm.  An integrable shallow water equation with peaked solitons.  Phys. Rev. Lett., 71(11):1661–1664, 1993.
  * [8] R. Camassa, D. D. Holm, and J. Hyman.  A new integrable shallow water equation.  Adv. Appl. Mech, 31:1–33, 1994.
  * [9] R. Camassa, J. Huang, and L. Lee.  On a completely integrable numerical scheme for a nonlinear shallow-water wave equation.  J. Nonlinear Math. Phys., 12(suppl. 1):146–162, 2005.
  * [10] R. Camassa, J. Huang, and L. Lee.  Integral and integrable algorithms for a nonlinear shallow-water wave equation.  J. Comput. Phys., 216(2):547–572, 2006.
  * [11] G. M. Coclite, H. Holden, and K. H. Karlsen.  Global weak solutions to a generalized hyperelastic-rod wave equation.  SIAM J. Math. Anal., 37(4):1044–1069 (electronic), 2005.
  * [12] G. M. Coclite, K. H. Karlsen, and N. H. Risebro.  A convergent finite difference scheme for the Camassa-Holm equation with general $H^{1}$ initial data.  Submitted, 2006.
  * [13] D. Cohen, B. Owren, and X. Raynaud.  Multi-symplectic integration of the Camassa-Holm equation.  Submitted, 2007.
  * [14] A. Constantin.  On the scattering problem for the Camassa-Holm equation.  R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457(2008):953–970, 2001.
  * [15] A. Constantin and J. Escher.  Global existence and blow-up for a shallow water equation.  Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26(2):303–328, 1998.
  * [16] A. Constantin and J. Escher.  Global weak solutions for a shallow water equation.  Indiana Univ. Math. J., 47(4):1527–1545, 1998.
  * [17] A. Constantin and B. Kolev.  On the geometric approach to the motion of inertial mechanical systems.  J. Phys. A, 35(32):R51–R79, 2002.
  * [18] A. Constantin and B. Kolev.  Geodesic flow on the diffeomorphism group of the circle.  Comment. Math. Helv., 78(4):787–804, 2003.
  * [19] A. Constantin and H. P. McKean.  A shallow water equation on the circle.  Comm. Pure Appl. Math., 52(8):949–982, 1999.
  * [20] A. Constantin and L. Molinet.  Global weak solutions for a shallow water equation.  Comm. Math. Phys., 211(1):45–61, 2000.
  * [21] H. H. Dai.  Model equations for nonlinear dispersive waves in a compressible mooney–rivlin rod.  Acta Mechanica, 127:193–207, 1998.
  * [22] R. Danchin.  A few remarks on the Camassa-Holm equation.  Differential Integral Equations, 14(8):953–988, 2001.
  * [23] R. Danchin.  A note on well-posedness for Camassa-Holm equation.  J. Differential Equations, 192(2):429–444, 2003.
  * [24] E. Feireisl.  Dynamics of viscous compressible fluids, volume 26 of Oxford Lecture Series in Mathematics and its Applications.  Oxford University Press, Oxford, 2004.
  * [25] O. B. Fringer and D. D. Holm.  Integrable vs. nonintegrable geodesic soliton behavior.  Phys. D, 150(3-4):237–263, 2001.
  * [26] B. Fuchssteiner and A. S. Fokas.  Symplectic structures, their Bäcklund transformations and hereditary symmetries.  Phys. D, 4(1):47–66, 1981/82.
  * [27] H. Holden and X. Raynaud.  Convergence of a finite difference scheme for the Camassa-Holm equation.  SIAM J. Num. Anal., 44(4):1655—1680, 2006.
  * [28] H. Holden and X. Raynaud.  A convergent numerical scheme for the Camassa-Holm equation based on multipeakons.  Discrete Contin. Dyn. Syst., 14(3):505–523, 2006.
  * [29] H. Holden and X. Raynaud.  Global conservative solutions of the Camassa-Holm equation - a Lagrangian point of view.  Comm. Partial Differential Equations, to appear.
  * [30] D. D. Holm, J. E. Marsden, and T. S. Ratiu.  The Euler-Poincaré equations and semidirect products with applications to continuum theories.  Adv. Math., 137(1):1–81, 1998.
  * [31] D. D. Holm and M. F. Staley.  Wave structure and nonlinear balances in a family of evolutionary PDEs.  SIAM J. Appl. Dyn. Syst., 2(3):323–380 (electronic), 2003.
  * [32] R. S. Johnson.  Camassa-Holm, Korteweg-de Vries and related models for water waves.  J. Fluid Mech., 455:63–82, 2002.
  * [33] H. Kalisch and X. Raynaud.  Convergence of a spectral projection of the Camassa-Holm equation.  Numer. Methods Partial Differential Equations, 22(5):1197–1215, 2006.
  * [34] Y. A. Li and P. J. Olver.  Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation.  J. Differential Equations, 162(1):27–63, 2000.
  * [35] H. Lundmark.  Formation and dynamics of shock waves in the Degasperis-Procesi equation.  J. Nonlinear Sci., 17(3):169–198, 2007.
  * [36] G. Misiołek.  A shallow water equation as a geodesic flow on the Bott-Virasoro group.  J. Geom. Phys., 24(3):203–208, 1998.
  * [37] G. Rodríguez-Blanco.  On the Cauchy problem for the Camassa-Holm equation.  Nonlinear Anal., 46(3, Ser. A: Theory Methods):309–327, 2001.
  * [38] J. Smoller.  Shock waves and reaction-diffusion equations, volume 258 of Grundlehren der Mathematischen Wissenschaften.  Springer-Verlag, New York, second edition, 1994.
  * [39] Z. Xin and P. Zhang.  On the weak solutions to a shallow water equation.  Comm. Pure Appl. Math., 53(11):1411–1433, 2000.
  * [40] Z. Xin and P. Zhang.  On the uniqueness and large time behavior of the weak solutions to a shallow water equation.  Comm. Partial Differential Equations, 27(9-10):1815–1844, 2002\.
  * [41] Y. Xu and C.-W. Shu.  A local discontinuous Galerkin method for the Camassa-Holm equation.  Siam J. Numer. Anal., to appear.