# Semi-implicit energy-preserving numerical schemes for stochastic wave
equation via SAV approach

Jianbo Cui11footnotemark: 1, Jialin Hong22footnotemark: 2, and Liying
Sun33footnotemark: 3,
11footnotemark: 1 Department of Applied Mathematics, The Hong Kong Polytechnic
University, Hung Hom, Hong Kong.
22footnotemark: 2 33footnotemark: 3 Institute of Computational Mathematics and
Scientific/Engineering Computing,
Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing 100190, P.R.China
33footnotemark: 3 Institute of Applied Physics and Computational Mathematics,
Beijing, 100094, China.

###### Abstract

In this paper, we propose and analyze semi-implicit numerical schemes for the
stochastic wave equation (SWE) with general nonlinearity and multiplicative
noise. These numerical schemes, called stochastic scalar auxiliary variable
(SAV) schemes, are constructed by transforming the considered SWE into a
higher dimensional stochastic system with a stochastic SAV. We prove that they
can be solved explicitly and preserve the modified energy evolution law and
the regularity structure of the original system. These structure-preserving
properties are the keys to overcoming the mutual effect of the noise and
nonlinearity. By proving new regularity estimates of the introduced SAV, we
establish the strong convergence rate of stochastic SAV schemes and the
further fully-discrete schemes with the finite element method in spatial
direction. To the best of our knowledge, this is the first result on the
construction and strong convergence of semi-implicit energy-preserving schemes
for nonlinear SWE.

AMS subject classification: 60H08, 60H35, 65C30.

Key Words:  stochastic wave equation, semi-implicit numerical scheme, energy
evolution law, SAV approach, strong convergence

$$$$footnotetext: 33footnotemark: 3Corresponding author:
<EMAIL_ADDRESS>
##  1 Introduction

In recent years, the stochastic wave equation has been widely exploited to
characterize the sound propagation in the sea, the dynamics of the primary
current density vector field within the grey matter of the human brain, heat
conduction around a ring, the dilatation of shock waves throughout the sun,
the vibration of a string under the action of stochastic forces, etc., (see,
e.g., [16, 17, 18, 23, 26]). As an intrinsic property of such stochastic
partial differential equation (SPDE), the energy evolution law describes the
longtime behaviour of the original system and is a keystone to prove the well-
posedness of SWE with complex nonlinearities (see, e.g., [4, 7, 8]). Due to
the loss of the explicit expression of the analytical solution, a lot of
researchers have been concerned with numerical schemes to simulate SWE and the
associated energy evolution law.

Up to now, various energy-preserving schemes have been developed for solving
SWEs (see, e.g., [1, 2, 5, 10, 13, 19]). For instance, the finite element
method and the stochastic trigonometric scheme are shown in [10] to preserve
the linear growth of the averaged energy for the linear SWE with additive
noise. With respect to the SWE with globally Lipschitz continuous coefficient
and additive noise, [2] presents that the discontinuous Galerkin finite
element method satisfies the trace formula. By means of the linear finite
element approximation and a stochastic trigonometric method, [1] designs a
fully-discrete scheme for the nonlinear SWE driven by multiplicative noise,
and proves that its numerical solution satisfies an almost trace formula in
additive noise case. [19] constructs several fully-discrete schemes which
preserve the averaged energy evolution law for nonlinear SWE driven by
multiplicative noise based on the Padé approximation. For the SWE with
superlinear coefficient and additive noise, [13] proposes an implicit fully-
discrete scheme by adopting the spectral Galerkin method and the averaged
vector field method, and establishes its strong convergence rate, where the
exponential integrability and energy-preserving property of the numerical
solution play key roles.

Despite fruitful results on energy-preserving numerical schemes for nonlinear
SWEs, one has to solve the large stochastic algebraic system by iterative
methods due to the implicitness of existing numerical schemes in general. In
practice, it would be ideal to be able to treat the nonlinear term in SWE
explicitly. However, it is known that the classical explicit and semi-implicit
numerical schemes for stochastic differential equations with superlinear
coefficients suffer from the strong or weak divergence phenomenon (see, e.g.,
[11, 20]). To deal with this issue, the existing approach often exploits
implicit schemes, or uses the tamed (or truncated) strategy to achieve
explicit numerical schemes which could not preserve the  energy evolution law.
Up to now, there has not been any numerical scheme which is explicitly
solvable and can preserve the averaged energy evolution law of nonlinear SWE.
One main objective of the present work is to fill this gap for nonlinear SWE
via a structure-preserving approach.

For deterministic conservative partial differential equations and gradient
flow systems, the SAV approach has achieved a lot of successes in constructing
semi-implicit numerical schemes which could preserve the energy decaying
property or the modified energy conservation law (see, e.g., [24, 25]). In
contrast, less result on the SAV schemes is known for SPDEs. A natural
question is how the noise influences the construction and analysis of
numerical schemes based on the SAV approach. It turns out that several new
challenges appear in this aspect for nonlinear SWE. First, the energy suffers
from the random effect and satisfies the stochastic energy evolution law,
rather than the energy conservation law. Second, the direct usage of SAV
approach may not balance the diffusion coefficient and the new auxiliary
variable, which brings troubles in constructing numerical schemes preserving
the modified energy evolution law. Last, due to the mutual effect of noise and
nonlinearity on the stochastic auxiliary variable, the strong convergence
analysis of stochastic SAV numerical schemes is more complicated than the
standard error estimate.

In this paper, we first transform the considered SWE into a higher dimensional
stochastic system (called the stochastic SAV reformulation) by introducing a
stochastic SAV. By further exploiting the exponential Runge–Kutta methods and
Runge–Kutta methods, as well as the spatial finite element method, we propose
two novel kinds of semi-linear temporal semi-discrete stochastic SAV schemes
and the further fully-discrete schemes. Then we prove that the proposed
stochastic SAV schemes could preserve the averaged energy evolution law for
the stochastic SAV reformulation, and overcome the divergence issue in the
superlinear coefficient case. By establishing new regularity estimates of the
stochastic auxiliary variable, we prove that the temporal strong convergence
order of stochastic SAV numerical schemes is $1$ under the globally Lipschitz
continuous condition. In particular, this result is even new for the
deterministic SWE. The convergence analysis of stochastic SAV numerical
schemes in the non-globally Lipschitz case is beyond the scope of this paper,
and will be studied in the future.

The rest of this paper is organized as follows. Section 2 presents an abstract
formulation of SWE, and introduces its fundamental properties. In Section 3,
the semi-implicit energy-preserving numerical schemes based on the SAV
approach for SWEs are proposed. Section 4 is devoted to deducing the strong
error estimate for the proposed numerical schemes. In Section 5, we present
energy-preserving fully-discrete schemes based on the SAV approach and the
finite element method. Numerical experiments are carried out in Section 6 to
verify theoretical results.

##  2 Stochastic wave equation

In this section, we present the main assumptions and introduce the properties
of the SWE with general nonlinearities and multiplicative noise, i.e.,

$\left\\{\begin{aligned}
&du(t)=v(t)dt,\quad\quad\;\;&\text{in}\;\;\mathcal{O}\times(0,T],\\\
&dv(t)=\Lambda u(t)dt-f(u(t))dt+g(\Theta
u(t))dW(t),\quad\quad\;\;&\text{in}\;\;\mathcal{O}\times(0,T],\\\
&u(0)=u_{0},\quad
v(0)=v_{0},\quad\;\;&\text{in}\;\;\mathcal{O},\end{aligned}\right.$ (1)

where $\mathcal{O}\in\mathbb{R}^{d},$ $d\leq 3,$ is a bounded convex domain
with polygonal boundary $\partial\mathcal{O}$, $T\in(0,\infty),$
$u_{0},v_{0}:\mathcal{O}\rightarrow\mathbb{R}$ are
$\mathcal{F}_{0}$-measurable, and $\Theta
u(t):=(u(t),\partial_{x_{1}}u(t),\cdots,\partial_{x_{d}}u(t))$ with
$\partial_{x_{i}}=\frac{\partial}{\partial x_{i}},i\leq d.$ Here,
$\Lambda=\sum\limits_{i=1}^{d}\frac{\partial^{2}}{\partial x_{i}^{2}}$ is the
Laplace operator with the homogeneous Dirichlet boundary condition with
eigenpairs $\\{(\lambda_{i},e_{i})\\}_{i=1}^{\infty},$ where
$0<\lambda_{1}\leq\lambda_{2}\leq\cdots,$ and $\\{e_{i}\\}_{i=1}^{\infty}$
forms an orthonormal basis in $L^{2}:=L^{2}(\mathcal{O};\mathbb{R}).$
Moreover,
$W(\cdot)=\sum\limits_{k=1}^{\infty}\mathbf{Q}^{\frac{1}{2}}e_{k}\beta_{k}(\cdot)$
is an $L^{2}$-valued $\mathbf{Q}$-Wiener process with respect to a filtered
probability space $(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathbb{P})$, where $\\{\beta_{k}\\}_{k\in\mathbb{N}^{+}}$ is a sequence of
i.i.d. real-valued Brownian motions and $\mathbf{Q}$ is a symmetric, positive
definite and finite trace operator. Denoting
$L^{p}:=L^{p}(\mathcal{O},\mathbb{R}),$ $p\geq 1,$ and
$\dot{\mathbb{H}}^{r}:=\mathcal{D}((-\Lambda)^{\frac{r}{2}}),$
$r\in\mathbb{R},$ equipped with the inner product $\langle
x,y\rangle_{\dot{\mathbb{H}}^{r}}=\left\langle(-\Lambda)^{\frac{r}{2}}x,(-\Lambda)^{\frac{r}{2}}y\right\rangle_{L^{2}}=\sum\limits_{i=1}^{\infty}\lambda_{i}^{r}\langle
x,e_{i}\rangle_{L^{2}}$$\langle y,e_{i}\rangle_{L^{2}},$ we introduce the mild
assumptions on $f:\dot{\mathbb{H}}^{1}\to\dot{\mathbb{H}}$ and
$g:\dot{\mathbb{H}}^{1}\otimes\dot{\mathbb{H}}^{\otimes
d}\to\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})$,
where $\dot{\mathbb{H}}:=\dot{\mathbb{H}}^{0}=L^{2}$ and
$\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})$
is the space of Hilbert–Schmidt operators from
$\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}})$ into $\dot{\mathbb{H}}$.

######  Assumption 2.1.

Assume that
$\|\mathbf{Q}^{\frac{1}{2}}\|_{\mathcal{L}_{2}(\dot{\mathbb{H}},\dot{\mathbb{H}})}<\infty,$

$\displaystyle\|f(u)\|_{\dot{\mathbb{H}}}+\|g(\Theta
u)\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\leq
c_{0}(\|u\|_{\dot{\mathbb{H}}^{1}}+1),$
$\displaystyle\|f(u)-f(\widetilde{u})\|_{\dot{\mathbb{H}}}+\|g(\Theta
u)-g(\Theta\widetilde{u})\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\leq
c_{1}\|u-\widetilde{u}\|_{\dot{\mathbb{H}}^{1}}$

for $u,\widetilde{u}\in\dot{\mathbb{H}}^{1},$ where $c_{0},c_{1}\geq 0$.

######  Assumption 2.2.

Assume that

$\displaystyle\|f(u)\|_{\dot{\mathbb{H}}}\leq
b_{1}(\|u\|_{\dot{\mathbb{H}}^{1}}),$
$\displaystyle\|f(u)-f(\widetilde{u})\|_{\dot{\mathbb{H}}}\leq
b_{2}(\|u\|_{\dot{\mathbb{H}}^{1}},\|\widetilde{u}\|_{\dot{\mathbb{H}}^{1}})\|u-\widetilde{u}\|_{\dot{\mathbb{H}}^{1}},$
$\displaystyle\|g(\Theta
u)\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\leq
c_{2}(1+\|u\|_{L^{2k+2}}^{k+1}+\|u\|_{\dot{\mathbb{H}}^{1}}),$
$\displaystyle\|g(\Theta
u)-g(\Theta\widetilde{u})\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\leq
b_{3}(\|u\|_{\dot{\mathbb{H}}^{1}},\|\widetilde{u}\|_{\dot{\mathbb{H}}^{1}})\|u-\widetilde{u}\|_{\dot{\mathbb{H}}^{1}},$

where $c_{2}\geq 0$, and $b_{1},b_{2},b_{3}$ are polynomials with degree
$2k+1,$ $2k$ and $k$, $k\in\mathbb{N}^{+},$ respectively. Furthermore, suppose
that
$\|\mathbf{Q}^{\frac{1}{2}}\|_{\mathcal{L}_{2}(\dot{\mathbb{H}},\dot{\mathbb{H}})}<\infty,$
and that there exist positive constants $c_{3},c_{4},\delta_{0}>0$ such that
the potential $F(u)=\int_{\mathcal{O}}\widetilde{F}(u(x))dx$ satisfies

$\displaystyle f(u)=\widetilde{F}^{\prime}(u)\;\quad\text{and}\;\quad
F(u)+\delta_{0}\geq
c_{3}\|u\|_{\dot{\mathbb{H}}}^{2}+c_{4}\|u\|_{L^{2k+2}}^{2k+2}.$

######  Remark 2.1.

A typical example of $f$ satisfying Assumption 2.2 is the Nemytskii operator
of $f(\xi)=\sum_{j=1}^{2k+1}a_{j}\xi^{j}$ with odd degree $2k+1$ and
$a_{2k+1}>0$. More precisely, when $d\leq 2,$ $k\in\mathbb{N}^{+}$ and when
$d=3,$ $k=1$ (see, e.g., [7]). It can be also found that Assumption 2.1 is a
special case of Assumption 2.2.

###  2.1 Well-posedness and energy evolution law

To present spatial and temporal regularity estimates, as well as the averaged
energy evolution law, of (LABEL:mod;swe), we let $X=(u,v)^{\top}$. Then the
compact form of (LABEL:mod;swe) reads

$dX(t)=AX(t)dt+\mathbb{F}(X(t))dt+\mathbb{G}(X(t))dW(t),\quad t\in(0,T],$ (2)

where

$\displaystyle X(0)=X_{0}=\begin{bmatrix}u_{0}\\\ v_{0}\end{bmatrix},\quad
A=\begin{bmatrix}0&I\\\
\Lambda&0\end{bmatrix},\quad\mathbb{F}(X(t))=\begin{bmatrix}0\\\
-f(u(t))\end{bmatrix},\quad\mathbb{G}=\begin{bmatrix}0\\\ g(\Theta
u(t))\end{bmatrix}$

with $I$ being the identity operator defined in $L^{2}$. Set the product space
$\mathbb{H}^{r}:=\dot{\mathbb{H}}^{r}\times\dot{\mathbb{H}}^{r-1},$
$r\in\mathbb{R},$ endowed with the inner product $\langle
X_{1},X_{2}\rangle_{\mathbb{H}^{r}}=\langle
x_{1},x_{2}\rangle_{\dot{\mathbb{H}}^{r}}+\langle
y_{1},y_{2}\rangle_{\dot{\mathbb{H}}^{r-1}}$ for any
$X_{1}=(x_{1},y_{1})^{\top}\in\mathbb{H}^{r}$ and
$X_{2}=(x_{2},y_{2})^{\top}\in\mathbb{H}^{r}.$ It can been shown that the
domain of operator $A$ is given by

$\mathcal{D}(A)=\left\\{X\in\mathbb{H}:AX=[v,\Lambda
u]^{\top}\in\mathbb{H}:=L^{2}\times\dot{\mathbb{H}}^{-1}\right\\}=\dot{\mathbb{H}}^{1}\times
L^{2},$

and $A$ generates a unitary group $E(t)=\exp(tA)=\begin{aligned}
\begin{bmatrix}\mathcal{C}(t)&(-\Lambda)^{-\frac{1}{2}}\mathcal{S}(t)\\\
-(-\Lambda)^{\frac{1}{2}}\mathcal{S}(t)&\mathcal{C}(t)\end{bmatrix}\end{aligned},$
$t\in\mathbb{R},$ where $\mathcal{C}(t)=\cos(t(-\Lambda)^{\frac{1}{2}})$ and
$\mathcal{S}(t)=\sin(t(-\Lambda)^{\frac{1}{2}})$ are cosine and sine
operators, respectively.

Under Assumption 2.2, one could follow the proof of [7, Lemma 4.1 and Theorem
4.2] and get the global existence of the mild solution.

######  Lemma 2.1 (Existence and uniqueness of mild solution).

Let Assumption 2.2 hold and $X_{0}\in\mathbb{H}^{1}$. Then there exists a
unique mild solution of (2), i.e.,

$\displaystyle
X(t)=E(t)X_{0}+\int_{0}^{t}E(t-s)\mathbb{F}(X(s))ds+\int_{0}^{t}E(t-s)\mathbb{G}(X(s))dW(s)$
(3)

for $t\in[0,T].$ Moreover, for any $p\in[2,\infty),$ there exists
$C(p,X_{0},\mathbf{Q},T)>0$ such that

$\displaystyle\sup_{t\in[0,T]}\|X(t)\|_{L^{p}(\Omega;\mathbb{H}^{1})}\leq
C(p,X_{0},\mathbf{Q},T).$

The Lyapunov energy functional $V_{1}:\mathbb{H}^{1}\rightarrow\mathbb{R}$ of
(2) is defined by

$\displaystyle
V_{1}(u,v)=\frac{1}{2}\|u\|^{2}_{\dot{\mathbb{H}}^{1}}+\frac{1}{2}\|v\|^{2}_{L^{2}}+\int_{\mathcal{O}}\widetilde{F}(u)dx,$
(4)

and plays a crucial role in the study of  well-posedness and blow-up of (2).
The following proposition introduces the averaged energy evolution law.

######  Proposition 2.1.

Let Assumption 2.2 hold and $X_{0}\in\mathbb{H}^{1}$. Then for $t\in[0,T],$
(2) admits

$\displaystyle\mathbb{E}[V_{1}(u(t),v(t))]=$
$\displaystyle\mathbb{E}[V_{1}(u_{0},v_{0})]+\frac{1}{2}\int_{0}^{t}\mathbb{E}\Big{[}{\rm{Tr}}(g(\Theta
u(s))\mathbf{Q}^{\frac{1}{2}}(g(\Theta
u(s))\mathbf{Q}^{\frac{1}{2}})^{*})\Big{]}ds,$ (5)

where ${\rm{Tr}}(g(\Theta u)\mathbf{Q}^{\frac{1}{2}}(g(\Theta
u)\mathbf{Q}^{\frac{1}{2}})^{*})=\sum\limits_{i=1}^{\infty}\langle g(\Theta
u)\mathbf{Q}^{\frac{1}{2}}e_{i},g(\Theta
u)\mathbf{Q}^{\frac{1}{2}}e_{i}\rangle_{L^{2}}$.

###  2.2 Regularity estimates

In this part, we provide the properties of $E(t),$ $t\in\mathbb{R},$ and the
regularity estimates of the mild solution of (2). The following lemma concerns
with the temporal Hölder continuity of both sine and cosine operators, which
have been discussed, for example, in [1].

######  Lemma 2.2.

For $\gamma\in[0,1],$ there exists a positive constant $C(\gamma)$ depending
on $\gamma$, such that

$\displaystyle\|(\mathcal{S}(t)-\mathcal{S}(s))(-\Lambda)^{-\frac{\gamma}{2}}\|_{\mathcal{L}(\dot{\mathbb{H}})}\leq
C(\gamma)(t-s)^{\gamma},$
$\displaystyle\|(\mathcal{C}(t)-\mathcal{C}(s))(-\Lambda)^{-\frac{\gamma}{2}}\|_{\mathcal{L}(\dot{\mathbb{H}})}\leq
C(\gamma)(t-s)^{\gamma}$

and
$\|(E(t)-E(s))X\|_{\mathbb{H}}\leq{\color[rgb]{0,0,0}C(\gamma)}(t-s)^{\gamma}\|X\|_{\mathbb{H}^{\gamma}}$
for all $t\geq s\geq 0,$ where $\mathcal{L}(\dot{\mathbb{H}})$ denotes the
space of all linear bounded operators from $\dot{\mathbb{H}}$ to
$\dot{\mathbb{H}}$.

######  Lemma 2.3.

For any $t\geq 0,$ $\mathcal{C}(t)$ and $\mathcal{S}(t)$ satisfy a
trigonometric identity in the sense that
$\|\mathcal{S}(t)x\|_{L^{2}}^{2}+\|\mathcal{C}(t)x\|_{L^{2}}^{2}=\|x\|_{L^{2}}^{2}$
for $x\in L^{2}.$

Based on the above trigonometric identity, it can be shown that
$\|E(t)\|_{\mathcal{L}(\mathbb{H})}=1,$ $t\in\mathbb{R}.$ By means of Lemmas
2.1-2.3, one could follow the similar arguments as in the proof of [13, Lemma
3.3] or [15, Theorem 7.4], and obtain the following regularity estimates of
the mild solution.

######  Proposition 2.2.

Let Assumption 2.2 hold and $T>0$. Assume that $X_{0}\in\mathbb{H}^{\beta}$
for some $\beta\geq 1$ and that

$\displaystyle\|(-\Lambda)^{\frac{\beta-1}{2}}f(u)\|_{L^{2}}+\|(-\Lambda)^{\frac{\beta-1}{2}}g(\Theta
u)\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\leq
b_{4}(\|u\|_{\dot{\mathbb{H}}^{1}},\|u\|_{\dot{\mathbb{H}}^{\beta-1}}),\;u\in\dot{\mathbb{H}}^{\beta},$

where $b_{4}$ is a polynomial. Then it holds that for any $p\geq 2,$

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}\Big{[}\|X(t)\|_{\mathbb{H}^{\beta}}^{p}\Big{]}\leq
C(p,X_{0},\mathbf{Q},T),$

and for $0\leq s\leq t\leq T,$

$\displaystyle\mathbb{E}\Big{[}\|u(t)-u(s)\|_{L^{2}}^{p}\Big{]}\leq
C(p,X_{0},\mathbf{Q},T)|t-s|^{p},$
$\displaystyle\mathbb{E}\Big{[}\|v(t)-v(s)\|_{\dot{\mathbb{H}}^{-1}}^{p}\Big{]}\leq
C(p,X_{0},\mathbf{Q},T)|t-s|^{\frac{p}{2}}.$

where $C(p,X_{0},\mathbf{Q},T)$ is a positive constant depending on
$p,X_{0},\mathbf{Q}$ and $T$.

Due to the loss of explicit expression of the exact solution, designing
numerical schemes for nonlinear stochastic differential equations  has become
an active area of research (see, e.g., [6, 12, 14, 21]). Below we shall focus
on the construction and analysis of the temporal and fully-discrete numerical
schemes for the considered SWE.

##  3 SAV approach for stochastic wave equation

The existing structure-preserving numerical schemes for the nonlinear SWE are
implicit in general. When implementing such numerical schemes, one has to
solve pathwisely complex stochastic algebraic systems, which often require a
lot of computational costs. To avoid such issue, we introduce the equivalent
form of (LABEL:mod;swe) with an auxiliary variable and construct efficient
structure-preserving numerical schemes in this section.

###  3.1 Stochastic wave equation with auxiliary variable

In this part, we introduce a scalar auxiliary variable
$q:=\sqrt{F(u)+\delta_{0}}.$ Here, $\delta_{0}$ is a constant to make $q$
well-posed, i.e., $F(u)+\delta_{0}\geq c>0$ for any
$u\in\dot{\mathbb{H}}^{1}$.  Then (LABEL:mod;swe) can be reformulated into the
following equivalent form

$\left\\{\begin{aligned} &du(t)=v(t)dt,\quad\;\;&\text{in}\;\;(0,T],\\\
&dv(t)=\Lambda u(t)dt-\frac{f(u(t))}{\sqrt{F(u(t))+\delta_{0}}}q(t)dt+g(\Theta
u(t))dW(t),\quad\;\;&\text{in}\;\;(0,T],\\\ &dq(t)=\frac{\langle
f(u(t)),\partial_{t}u(t)\rangle_{L^{2}}}{2\sqrt{F(u(t))+\delta_{0}}}dt,\;\;&\text{in}\;\;(0,T].\end{aligned}\right.$
(6)

######  Lemma 3.1.

Let Assumption 2.2 hold and $X_{0}\in\mathbb{H}^{1}$. Then the averaged energy
evolution law of
$V(u(t),v(t),q(t)):=\frac{1}{2}\|u(t)\|^{2}_{\dot{\mathbb{H}}^{1}}+\frac{1}{2}\|v(t)\|^{2}_{L^{2}}+q(t)$
has the following form

$\displaystyle\mathbb{E}\Big{[}V(u(t),v(t),q(t))\Big{]}$ $\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}V(u(0),v(0),q(0))\Big{]}+\frac{1}{2}\int_{0}^{t}\mathbb{E}\Big{[}{\rm{Tr}}(g(\Theta
u(s))\mathbf{Q}^{\frac{1}{2}}(g(\Theta
u(s))\mathbf{Q}^{\frac{1}{2}})^{*})\Big{]}ds,$ (7)

where $t\in{[0,T]}.$ Furthermore, it holds that for $p\geq 2$ and $0\leq s\leq
t\leq T,$

$\displaystyle\mathbb{E}\Big{[}|q(t)-q(s)|^{p}\Big{]}$ $\displaystyle\leq
C(p,X_{0},\mathbf{Q},T)|t-s|^{p},$

where $C(p,X_{0},\mathbf{Q},T)$ is a positive constant depending on
$p,X_{0},\mathbf{Q},$ and $T$.

###### Proof.

Proposition 2.1 implies the averaged evolution law (3.1). Notice that

$q(t)-q(s)=\int_{s}^{t}\frac{\langle
f(u(r)),v(r)\rangle_{L^{2}}}{\sqrt{F(u(r))+\delta_{0}}}dr.$

By using  Assumption 2.2, Hölder’s inequality and Lemma 2.1, we have

$\displaystyle\mathbb{E}\Big{[}|q(t)-q(s)|^{p}\Big{]}$
$\displaystyle\leq\frac{1}{c^{p}}\int_{s}^{t}\mathbb{E}\Big{[}b_{1}^{p}(\|u(r)\|_{\dot{\mathbb{H}}^{1}})\|v(r)\|_{L^{2}}^{p}\Big{]}dr\leq
C(p,X_{0},\mathbf{Q},T)|t-s|^{p},$

where $b_{1}$ is a polynomial. ∎

###  3.2 Semi-implicit stochastic SAV  schemes

Let $\tau$ be the time step-size, and denote $t_{n}=n\tau$ for
${\color[rgb]{0,0,0}n\in\\{1,\ldots,N\\}}$ with $t_{N}=T.$ Let $u_{n}$ be the
numerical approximation of $u(t_{n})$ for $n\in\\{1,\ldots,N\\}.$ We propose
two novel kinds of semi-implicit energy-preserving numerical schemes via the
SAV approach. In the rest of this paper, the constant $C$ may be different
from line to line but never depending on $N$ and $\tau$.

The first natural choice is based on the Runge–Kutta type discretization and
the SAV approach.  To make it clear, we employ the midpoint  scheme to
approximate the linear unbounded part of stochastic system (6) and obtain the
midpoint SAV  scheme, i.e.,

$\displaystyle
u_{n+1}=u_{n}+\frac{\tau}{2}(v_{n}+v_{n+1})+\frac{\tau}{2}g(\Theta
u_{n})\delta W_{n},$ (8) $\displaystyle
v_{n+1}=v_{n}+\frac{\tau}{2}\Lambda(u_{n}+u_{n+1})-\tau\frac{f(\widehat{u}_{n})}{\sqrt{F(\widehat{u}_{n})+\delta_{0}}}\frac{q_{n}+q_{n+1}}{2}+g(\Theta
u_{n})\delta W_{n},$ $\displaystyle q_{n+1}=q_{n}+\frac{\langle
f(\widehat{u}_{n}),u_{n+1}-u_{n}\rangle_{L^{2}}}{2\sqrt{F(\widehat{u}_{n})+\delta_{0}}}.$

Here, $\widehat{u}_{n}=u_{n}$ (or $\frac{3u_{n}-u_{n-1}}{2}$ with
$u_{-1}=u_{0}$) , $n\in\\{0,1,\ldots,N-1\\},$ and $\delta
W_{n}=W(t_{n+1})-W(t_{n}).$ We would like to remark that the modified term
$\frac{\tau}{2}g(\Theta u_{n})\delta W_{n}$ is used to balance the diffusion
coefficient and the stochastic auxiliary variable. Without this modification,
the direct discretization of (6) fails to preserve the modified energy
evolution law.

An alternative way to construct semi-linear stochastic SAV  schemes is
inspired by the phenomenon that the exponential Runge–Kutta  schemes often
require less restriction for SPDEs (see, e.g., [1, 3, 9]) with regard to
optimal strong convergence rate. By employing the exponential Euler  scheme to
linear unbounded part of stochastic system (6), we get

$\displaystyle
X_{n+1}=E(\tau)X_{n}+A^{-1}(E(\tau)-I)\Big{(}0,-\frac{f(\widehat{u}_{n})}{\sqrt{F(\widehat{u}_{n})+\delta_{0}}}\frac{q_{n}+q_{n+1}}{2}\Big{)}^{\top}$
(9) $\displaystyle\quad\quad\quad+E(\tau)(0,g(\Theta u_{n})\delta
W_{n})^{\top},$ $\displaystyle q_{n+1}=q_{n}+\frac{\langle
f(\widehat{u}_{n}),u_{n+1}-u_{n}\rangle_{L^{2}}}{2\sqrt{F(\widehat{u}_{n})+\delta_{0}}},$

where $X_{n+1}=(u_{n+1},v_{n+1})^{\top},$ $n\in\\{0,1,\ldots,N-1\\}.$ We would
like to mention that the proposed  schemes (8) and (9) have a great potential
for approximating other SPDEs, such as stochastic nonlinear Schrödinger
equations and parabolic SPDEs.

One main advantage of the proposed  schemes is that they can be solved
efficiently. We take (9) as an example to illustrate this fact. Notice that
(9) can be rewritten as

$\displaystyle u_{n+1}=$ $\displaystyle
e_{11}u_{n}+e_{12}v_{n}-a^{1}\frac{f(\widehat{u}_{n})}{\sqrt{F(\widehat{u}_{n})+\delta_{0}}}\frac{q_{n}+q_{n+1}}{2}+e_{12}g(\Theta
u_{n})\delta W_{n},$ (10) $\displaystyle v_{n+1}=$ $\displaystyle
e_{21}u_{n}+e_{22}v_{n}-a^{2}\frac{f(\widehat{u}_{n})}{\sqrt{F(\widehat{u}_{n})+\delta_{0}}}\frac{q_{n}+q_{n+1}}{2}+e_{22}g(\Theta
u_{n})\delta W_{n},$ (11) $\displaystyle q_{n+1}=$ $\displaystyle
q_{n}+\frac{\langle
f(\widehat{u}_{n}),u_{n+1}-u_{n}\rangle_{L^{2}}}{2\sqrt{F(\widehat{u}_{n})+\delta_{0}}}$
(12)

with $n\in\\{0,1,\ldots,N-1\\}$, $e_{11}=e_{22}=\mathcal{C}(\tau),$
$e_{12}=(-\Lambda)^{-\frac{1}{2}}\mathcal{S}(\tau),$
$e_{21}=-(-\Lambda)^{\frac{1}{2}}\mathcal{S}(\tau),$ $a^{1}=-(-\Lambda)^{-1}$
$(\mathcal{C}(\tau)-I)$ and
$a^{2}=(-\Lambda)^{-\frac{1}{2}}\mathcal{S}(\tau).$ By eliminating
$\frac{q_{n}+q_{n+1}}{2}$, we deduce

$\displaystyle u_{n+1}+\gamma_{n}\langle
f(\widehat{u}_{n}),u_{n+1}\rangle_{L^{2}}=\Gamma_{n},$ (13)

where
$\gamma_{n}=\frac{a^{1}f(\widehat{u}_{n})}{4(F(\widehat{u}_{n})+\delta_{0})}$
and

$\displaystyle\Gamma_{n}=e_{11}u_{n}+e_{12}v_{n}-a^{1}\frac{f(\widehat{u}_{n})}{\sqrt{F(\widehat{u}_{n})+\delta_{0}}}q_{n}+\gamma_{n}\langle
f(\widehat{u}_{n}),u_{n}\rangle_{L^{2}}+e_{12}g(\Theta u_{n})\delta W_{n}.$

Taking the inner product of (13) with $f(\widehat{u}_{n}),$ we derive

$\displaystyle\left(1+\langle
f(\widehat{u}_{n}),\gamma_{n}\rangle_{L^{2}}\right)\langle
f(\widehat{u}_{n}),u_{n+1}\rangle_{L^{2}}=\langle
f(\widehat{u}_{n}),\Gamma_{n}\rangle_{L^{2}}.$

Here $\langle f(\widehat{u}_{n}),\gamma_{n}\rangle_{L^{2}}\geq 0$ due to
$a^{1}=(-\Lambda)^{-1}(I-\mathcal{C}(\tau))$. Then it follows that

$\displaystyle\langle f(\widehat{u}_{n}),u_{n+1}\rangle_{L^{2}}=\frac{\langle
f(\widehat{u}_{n}),\Gamma_{n}\rangle_{L^{2}}}{1+\langle
f(\widehat{u}_{n}),\gamma_{n}\rangle_{L^{2}}}.$

After solving $\langle f(\widehat{u}_{n}),u_{n+1}\rangle_{L^{2}}$ from the
linear system, $u_{n+1}$ is then updated from (13). Subsequently, $q_{n+1}$ is
obtained from (12). Finally, we get $v_{n+1}$ from (11). Utilizing the similar
procedures, one can also verify that $u_{n+1},$ $q_{n+1}$ and $v_{n+1}$ in (8)
can be solved explicitly.

Next we will show that the proposed schemes could overcome the divergent issue
in the superlinear coefficient case and preserve the energy evolution law,
which, together with the regularity estimates of the discrete auxiliary
variable, plays the key role in the strong convergence analysis 4.

###  3.3  The preservation of energy evolution law

In this part, we address the important property of the proposed  schemes,
i.e., they could inherit the averaged energy evolution law of the SWE
(LABEL:mod;swe) (or (2)). The rigorous proof of these results requires some
regularization technique s, like taking advantage of the spectral Galerkin
approximation, such that the integration by parts formula makes sense, and
then taking limit. For simplicity, we omit these tedious procedures.

######  Proposition 3.1.

Let Assumption 2.2 hold and $X_{0}\in\mathbb{H}^{1}$. Numerical  schemes (8)
and (9) preserve the discrete averaged modified energy evolution law, for
$n\in\\{0,1,\ldots,N-1\\},$

$\displaystyle\mathbb{E}[V(u_{n+1},v_{n+1},q_{n+1})]=$
$\displaystyle\mathbb{E}[V(u_{n},v_{n},q_{n})]+\frac{\tau}{2}\mathbb{E}\Big{[}{\rm{Tr}}\left(g(\Theta
u_{n})\mathbf{Q}^{\frac{1}{2}}(g(\Theta
u_{n})\mathbf{Q}^{\frac{1}{2}})^{*}\right)\Big{]},$

where
$V(u_{n},v_{n},q_{n})=\frac{1}{2}\|u_{n}\|^{2}_{\dot{\mathbb{H}}^{1}}+\frac{1}{2}\|v_{n}\|^{2}_{L^{2}}+q_{n}^{2}.$

###### Proof.

To simplify the presentation, we only present the proof of (9) since that of
(8) is similar. Fix $n\in\\{0,1,\ldots,N-1\\},$ and denote
$M=\begin{bmatrix}-\Lambda&0\\\ 0&I\end{bmatrix},$ $Id=\begin{bmatrix}I&0\\\
0&I\end{bmatrix},$ $J=\begin{bmatrix}0&I\\\ -I&0\end{bmatrix},$
$\tilde{g}_{n}=\begin{bmatrix}0\\\ g(\Theta u_{n})\delta W_{n}\end{bmatrix}$
and $\tilde{f}_{n}=\begin{bmatrix}0\\\
-\frac{f(\widehat{u}_{n})}{\sqrt{F(\widehat{u}_{n})+\delta_{0}}}\frac{q_{n}+q_{n+1}}{2}\end{bmatrix}.$
Then using the integration by parts formula, it follows that

$\displaystyle\frac{1}{2}\|u_{n}\|_{\dot{\mathbb{H}}^{1}}^{2}+\frac{1}{2}\|v_{n}\|_{L^{2}}^{2}=\frac{1}{2}\int_{\mathcal{O}}X_{n}^{\top}MX_{n}dx.$
(14)

Notice that

$\displaystyle X_{n+1}^{\top}MX_{n+1}-X_{n}^{\top}MX_{n}$ $\displaystyle=$
$\displaystyle
X_{n}^{\top}(E(\tau)^{\top}ME(\tau)-M)X_{n}+2X_{n}^{\top}E(\tau)^{\top}M(E(\tau)-Id)A^{-1}\tilde{f}_{n}$
$\displaystyle+2X_{n}^{\top}E(\tau)^{\top}ME(\tau)\tilde{g}_{n}+(\tilde{f}_{n})^{\top}(A^{-1})^{\top}(E(\tau)-Id)^{\top}M(E(\tau)-Id)A^{-1}\tilde{f}_{n}$
$\displaystyle+2(\tilde{f}_{n})^{\top}(A^{-1})^{\top}(E(\tau)-Id)^{\top}ME(\tau)\tilde{g}_{n}+(\tilde{g}_{n})^{\top}E(\tau)^{\top}ME(\tau)\tilde{g}_{n},$

and

$\displaystyle q_{n+1}^{2}-q_{n}^{2}=\frac{\langle
f(\widehat{u}_{n}),u_{n+1}\rangle_{L^{2}}}{\sqrt{F(\widehat{u}_{n})+\delta_{0}}}\frac{q_{n}+q_{n+1}}{2}-\frac{\langle
f(\widehat{u}_{n}),u_{n}\rangle_{L^{2}}}{\sqrt{F(\widehat{u}_{n})+\delta_{0}}}\frac{q_{n}+q_{n+1}}{2}$
$\displaystyle=$
$\displaystyle\int_{\mathcal{O}}(X_{n})^{\top}(E(\tau)-Id)^{\top}J^{-1}\tilde{f}_{n}dx+\int_{\mathcal{O}}(\tilde{f}_{n})^{\top}(A^{-1})^{\top}(E(\tau)-Id)^{\top}J^{-1}\tilde{f}_{n}dx$
$\displaystyle+\int_{\mathcal{O}}(\tilde{g}_{n})^{\top}E(\tau)^{\top}J^{-1}\tilde{f}_{n}dx.$

As a consequence, due to $MA^{-1}=J^{-1}$ and the fact that
$\int_{\mathcal{O}}\tilde{f}_{n}^{\top}(A^{-1})^{\top}ME(\tau)\tilde{g}_{n}dx=\int_{\mathcal{O}}\tilde{g}_{n}^{\top}E(\tau)^{\top}MA^{-1}\tilde{f}_{n}dx$,
we obtain

$\displaystyle\frac{1}{2}\int_{\mathcal{O}}(X_{n+1}^{\top}MX_{n+1}-X_{n}^{\top}MX_{n})dx+q_{n+1}^{2}-q_{n}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathcal{O}}X_{n}^{\top}(E(\tau)^{\top}ME(\tau)-M)X_{n}dx+\frac{1}{2}\int_{\mathcal{O}}(\tilde{g}_{n})^{\top}E(\tau)^{\top}ME(\tau)\tilde{g}_{n}dx$
$\displaystyle+\int_{\mathcal{O}}X_{n}^{\top}\left(E(\tau)^{\top}M(E(\tau)-Id)A^{-1}+(E(\tau)-Id)^{\top}J^{-1}\right)\tilde{f}_{n}dx$
$\displaystyle+\int_{\mathcal{O}}X_{n}^{\top}E(\tau)^{\top}ME(\tau)\tilde{g}_{n}dx$
$\displaystyle+\frac{1}{2}\int_{\mathcal{O}}(\tilde{f}_{n})^{\top}(A^{-1})^{\top}(E(\tau)-Id)^{\top}M\left(E(\tau)+Id\right)A^{-1}\tilde{f}_{n}dx$
$\displaystyle+\int_{\mathcal{O}}(\tilde{f}_{n})^{\top}(A^{-1})^{\top}\left((E(\tau)-Id)^{\top}ME(\tau)+ME(\tau)\right)\tilde{g}_{n}dx.$

From $(\tilde{f}_{n})^{\top}(A^{-1})^{\top}M\tilde{g}_{n}=0$, it follows that

$\displaystyle\frac{1}{2}\int_{\mathcal{O}}(X_{n+1}^{\top}MX_{n+1}-X_{n}^{\top}MX_{n})dx+q_{n+1}^{2}-q_{n}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathcal{O}}(X_{n}+A^{-1}\tilde{f}+\tilde{g}_{n})^{\top}(E(\tau)^{\top}ME(\tau)-M)(X_{n}+A^{-1}\tilde{f}+\tilde{g}_{n})dx$
$\displaystyle+\frac{1}{2}\int_{\mathcal{O}}(\tilde{f}_{n})^{\top}(A^{-1})^{\top}(E(\tau)^{\top}M-ME(\tau))A^{-1}\tilde{f}_{n}dx$
(15)
$\displaystyle+\int_{\mathcal{O}}X_{n}^{\top}M\tilde{g}_{n}dx+\frac{1}{2}\int_{\mathcal{O}}(\tilde{g}_{n})^{\top}M\tilde{g}_{n}dx.$

According to the unitary property of $E(\tau)$ and the skew symmetry of the
matrix $E(\tau)^{\top}M-ME(\tau),$ taking expectation leads to

$\displaystyle\mathbb{E}(\frac{1}{2}\|u_{n+1}\|^{2}_{\dot{\mathbb{H}}^{1}}+\frac{1}{2}\|v_{n+1}\|^{2}_{L^{2}}+q_{n+1}^{2})$
$\displaystyle=$
$\displaystyle\mathbb{E}(\frac{1}{2}\|u_{n}\|^{2}_{\dot{\mathbb{H}}^{1}}+\frac{1}{2}\|v_{n}\|^{2}_{L^{2}}+q_{n}^{2})+\frac{\tau}{2}{\rm{Tr}}\left(g(\Theta
u_{n})\mathbf{Q}^{\frac{1}{2}}(g(\Theta
u_{n})\mathbf{Q}^{\frac{1}{2}})^{*}\right),$

which completes the proof. ∎

As a result of Proposition 3.1, we are in a position to present a priori
estimates of numerical solutions.

######  Corollary 3.1.

Let Assumption 2.2 hold and $X_{0}\in\mathbb{H}^{1}$. In addition suppose that
$\|g(\Theta
u)\|_{\mathcal{L}_{2}({\bf{Q}}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\leq
c_{2}(1+\|u\|_{\dot{\mathbb{H}}^{1}})$ with $c_{2}>0.$ Then it holds that for
any $p\geq 1$,

$\displaystyle\sup_{n\leq
N}\mathbb{E}\big{[}V^{p}(u_{n},v_{n},q_{n})\big{]}\leq
C(p,X_{0},\mathbf{Q},T),$

where $(u_{n},v_{n},q_{n})$ is the numerical solution of (8) or (9), and
$C(p,X_{0},\mathbf{Q},T)>0$.

###### Proof.

By (15), $\|g(\Theta
u)\|_{\mathcal{L}_{2}({\bf{Q}}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\leq
c_{2}(1+\|u\|_{\dot{\mathbb{H}}^{1}})$, the Burkhölder inequality, as well as
Hölder’s and Young’s inequalities, we get that for $p\geq 1,$

$\displaystyle\quad\mathbb{E}[V^{p}(u_{n+1},v_{n+1},q_{n+1})]$
$\displaystyle\leq\mathbb{E}[V^{p}(u_{n},v_{n},q_{n})](1+C(p)\tau)+p\mathbb{E}[V^{p-1}(u_{n},v_{n},q_{n})\langle
v_{n},g(\Theta u_{n})\delta W_{n}\rangle_{L^{2}}]$
$\displaystyle+C(p)\mathbb{E}\Big{[}\langle g(\Theta u_{n})\delta
W_{n},g(\Theta u_{n})\delta
W_{n}\rangle_{L^{2}}^{p}\Big{]}+C(p)\tau\mathbb{E}\Big{[}\|v_{n}\|_{L^{2}}^{2p}\Big{]}+C(p)\tau\mathbb{E}\Big{[}1+\|u_{n}\|_{\dot{\mathbb{H}}^{1}}^{2p}\Big{]}$
$\displaystyle\leq\mathbb{E}[V^{p}(u_{n},v_{n},q_{n})](1+C(p)\tau)+C(p)\tau\mathbb{E}\Big{[}\|v_{n}\|_{L^{2}}^{2p}\Big{]}+C(p)\tau\mathbb{E}\Big{[}1+\|u_{n}\|_{\dot{\mathbb{H}}^{1}}^{2p}\Big{]},$

where we have used the independent increment property of $W(\cdot)$ and
Proposition 3.1 in the last inequality. The discrete Grönwall’s inequality
yields the desired result. ∎

##  4 Strong convergence rate of stochastic SAV  schemes

In this section, we provide a generic approach to study the strong convergence
rates of the proposed stochastic SAV  schemes (8)-(9).

###  4.1 Properties of the discrete auxiliary variable

In the following, we first show a priori estimates of both $q_{n}$ and $u_{n}$
for ${n\in\\{1,\ldots,N\\}}$, which are of vital importance in the study of
strong error estimates of the proposed stochastic energy-preserving  schemes
via the SAV approach.

######  Lemma 4.1.

Under the condition of Corollary 3.1, it holds that

$\displaystyle{\sup\limits_{j\in\\{0,1,\ldots,N-1\\}}}\mathbb{E}[\|u_{j+1}-u_{j}\|_{L^{2}}^{p}]\leq
C\tau^{p},$

where $u_{j}$ is the numerical solution of (8) or (9),
$N\in\mathbb{N}^{+},N\tau=T,$ and $C:=C(p,X_{0},\mathbf{Q},T)>0$.

###### Proof.

Fix $j\in\\{0,1,\ldots,N-1\\}.$ Based on the properties that $|\cos(x)-1|\leq
C|x|^{2}$ and $|\frac{\sin(x)}{x}|\leq C$ for some $C>0,$ we have

$\displaystyle\|u_{j+1}-u_{j}\|_{L^{2}}$ $\displaystyle\leq$
$\displaystyle\|(\cos(\tau(-\Lambda)^{\frac{1}{2}}-I)u_{j}\|_{L^{2}}+\|(-\Lambda)^{-\frac{1}{2}}\sin(\tau(-\Lambda)^{\frac{1}{2}})v_{j}\|_{L^{2}}$
$\displaystyle+\|(-\Lambda)^{-1}(\cos(\tau(-\Lambda)^{\frac{1}{2}}-I)\frac{f(\widehat{u}_{j})}{\sqrt{F(\widehat{u}_{j})+\delta_{0}}}\frac{q_{j}+q_{j+1}}{2}\|_{L^{2}}$
$\displaystyle+\|(-\Lambda)^{-\frac{1}{2}}\sin(\tau(-\Lambda)^{\frac{1}{2}})g(\Theta
u_{j})\delta W_{j}\|_{L^{2}}$ $\displaystyle\leq$ $\displaystyle
C\tau(\|u_{j}\|_{\dot{\mathbb{H}}^{1}}+\|v_{j}\|_{L^{2}})+C\tau^{2}\frac{\|f(\widehat{u}_{j})\|_{L^{2}}}{\sqrt{F(\widehat{u}_{j})+\delta_{0}}}(|q_{j}|+|q_{j+1}|)+\tau\|g(\Theta
u_{j})\delta W_{j}\|_{L^{2}}.$

Thanks to $\widehat{u}_{j}=u_{j}$ or $\frac{3u_{j}-u_{j-1}}{2}$, taking the
$p$th moment, and using Corollary 3.1, Young’s and Hölder’s inequalities, we
obtain

$\displaystyle\quad\mathbb{E}\Big{[}\|u_{j+1}-u_{j}\|_{L^{2}}^{p}\Big{]}$
$\displaystyle\leq
C(p)\tau^{p}\Big{(}\mathbb{E}[\|u_{j}\|_{\dot{\mathbb{H}}^{1}}^{p}]+\mathbb{E}[\|v_{j}\|_{\dot{\mathbb{H}}}^{p}]+\mathbb{E}[b(\|u_{j}\|_{\dot{\mathbb{H}}^{1}},\|\widehat{u}_{j}\|_{\dot{\mathbb{H}}^{1}})]+\mathbb{E}[q_{j}^{2p}+q_{j+1}^{2p}]\Big{)}$
$\displaystyle\leq C(p,X_{0},\mathbf{Q},T)\tau^{p},$

where $b$ is a polynomial. This completes the proof. ∎

######  Proposition 4.1.

Let the condition of Corollary 3.1 hold and
$|q_{0}-\sqrt{F(\widehat{u}_{0})+\delta_{0}}|\leq C\tau$. Suppose that

$|\langle f^{\prime}(u)v,w\rangle|\leq
b_{5}(\|u\|_{\dot{\mathbb{H}}^{1}},\|v\|_{\dot{\mathbb{H}}^{1}},\|w\|_{\dot{\mathbb{H}}^{1}})\|v\|_{L^{2}}^{\gamma_{1}}\|w\|_{L^{2}}^{\gamma_{1}}$

for some $\gamma_{1}>\frac{1}{2}$ and some polynomial $b_{5}.$ Then for (8)
and (9), it holds that

$\displaystyle{\sup\limits_{j\in\\{1,\ldots,N\\}}}\mathbb{E}\Big{[}|\sqrt{F(\widehat{u}_{j})+\delta_{0}}-q_{j}|^{p}\Big{]}\leq
C\tau^{\min(1,2\gamma_{1}-1)p},$

where $C:=C(p,X_{0},\mathbf{Q},T)>0$, $N\in\mathbb{N}^{+},N\tau=T.$

###### Proof.

Fix $j\in\\{1,\ldots,N-1\\}.$ The definitions of $q_{j}$ and
$F(\widehat{u}_{j})$, and the Taylor expansion yield that

$\displaystyle\sqrt{F(\widehat{u}_{j+1})+\delta_{0}}-q_{j+1}$ $\displaystyle=$
$\displaystyle\sqrt{F(\widehat{u}_{j})+\delta_{0}}-q_{j}-\tau\frac{\langle
f(\widehat{u}_{j}),\frac{u_{j+1}-u_{j}}{\tau}\rangle_{L^{2}}}{2\sqrt{F(\widehat{u}_{j})+\delta_{0}}}+\frac{F(\widehat{u}_{j+1})-F(\widehat{u}_{j})}{\sqrt{F(\widehat{u}_{j+1})+\delta_{0}}+\sqrt{F(\widehat{u}_{j})+\delta_{0}}}$
$\displaystyle=$
$\displaystyle\sqrt{F(\widehat{u}_{j})+\delta_{0}}-q_{j}-\tau\frac{\langle
f(\widehat{u}_{j}),\frac{u_{j+1}-u_{j}}{\tau}\rangle_{L^{2}}}{2\sqrt{F(\widehat{u}_{j})+\delta_{0}}}+\tau\frac{\langle
f(\widehat{u}_{j}),\frac{\widehat{u}_{j+1}-\widehat{u}_{j}}{\tau}\rangle_{L^{2}}}{\sqrt{F(\widehat{u}_{j})+\delta_{0}}+\sqrt{F(\widehat{u}_{j+1})+\delta_{0}}}$
$\displaystyle+\tau\frac{\langle\int_{0}^{1}f^{\prime}(\widehat{u}_{j}+\theta(\widehat{u}_{j+1}-\widehat{u}_{j})d\theta(\widehat{u}_{j+1}-\widehat{u}_{j}),\frac{\widehat{u}_{j+1}-\widehat{u}_{j}}{\tau}\rangle_{L^{2}}}{\sqrt{F(\widehat{u}_{j})+\delta_{0}}+\sqrt{F(\widehat{u}_{j+1})+\delta_{0}}}.$

By applying Hölder’s inequality, we obtain

$\displaystyle|\sqrt{F(\widehat{u}_{j+1})+\delta_{0}}-q_{j+1}|$
$\displaystyle\leq$
$\displaystyle|\sqrt{F(\widehat{u}_{j})+\delta_{0}}-q_{j}|$
$\displaystyle+\tau|\langle
f(\widehat{u}_{j}),\frac{\widehat{u}_{j+1}-\widehat{u}_{j}}{\tau}\rangle_{L^{2}}|\Big{|}\frac{\sqrt{F(\widehat{u}_{j+1})+\delta_{0}}-\sqrt{F(\widehat{u}_{j})+\delta_{0}}}{2(F(\widehat{u}_{j})+\delta_{0})+2\sqrt{F(\widehat{u}_{j})+\delta_{0}}\sqrt{F(\widehat{u}_{j+1})+\delta_{0}}}\Big{|}$
$\displaystyle+\tau\frac{\Big{|}\langle\int_{0}^{1}f^{\prime}(\widehat{u}_{j}+\theta(\widehat{u}_{j+1}-\widehat{u}_{j})d\theta(\widehat{u}_{j+1}-\widehat{u}_{j}),\frac{\widehat{u}_{j+1}-\widehat{u}_{j}}{\tau}\rangle_{L^{2}}\Big{|}}{\sqrt{F(\widehat{u}_{j})+\delta_{0}}+\sqrt{F(\widehat{u}_{j+1})+\delta_{0}}}$
$\displaystyle=:$
$\displaystyle|\sqrt{F(\widehat{u}_{j})+\delta_{0}}-q_{j}|+A_{1}+A_{2}.$

Due to the assumption of $f,$ there exists a polynomial $b$ such that

$\displaystyle A_{1}\leq$ $\displaystyle\tau|\langle
f(\widehat{u}_{j}),\frac{\widehat{u}_{j+1}-\widehat{u}_{j}}{\tau}\rangle_{L^{2}}|\Big{|}\frac{F(\widehat{u}_{j+1})-F(\widehat{u}_{j})}{2\sqrt{F(\widehat{u}_{j})+\delta_{0}}(\sqrt{F(\widehat{u}_{j+1})+\delta_{0}}+\sqrt{F(\widehat{u}_{j})+\delta_{0}})^{2}}\Big{|}$
$\displaystyle\leq$ $\displaystyle
Cb(\|\widehat{u}_{j}\|_{\dot{\mathbb{H}}^{1}},\|\widehat{u}_{j+1}\|_{\dot{\mathbb{H}}^{1}})\|\widehat{u}_{j+1}-\widehat{u}_{j}\|_{L^{2}}^{2}.$

For the term $A_{2},$ using $|\langle f^{\prime}(u)v,w\rangle|\leq
b_{5}(\|u\|_{\dot{\mathbb{H}}^{1}},\|v\|_{\dot{\mathbb{H}}^{1}},\|w\|_{\dot{\mathbb{H}}^{1}})\|v\|_{L^{2}}^{\gamma_{1}}\|w\|_{L^{2}}^{\gamma_{1}}$,
it holds that

$\displaystyle A_{2}\leq$ $\displaystyle
Cb_{5}(\|\widehat{u}_{j}\|_{\dot{\mathbb{H}}^{1}},\|\widehat{u}_{j+1}\|_{\dot{\mathbb{H}}^{1}})\|\widehat{u}_{j+1}-\widehat{u}_{j}\|_{L^{2}}^{2\gamma_{1}}.$

In sum,

$\displaystyle|\sqrt{F(\widehat{u}_{j+1})+\delta_{0}}-q_{j+1}|$
$\displaystyle\leq$
$\displaystyle|\sqrt{F(\widehat{u}_{j})+\delta_{0}}-q_{j}|+C\Big{(}b(\|\widehat{u}_{j}\|_{\dot{\mathbb{H}}^{1}},\|\widehat{u}_{j+1}\|_{\dot{\mathbb{H}}^{1}})+b_{5}(\|\widehat{u}_{j}\|_{\dot{\mathbb{H}}^{1}},\|\widehat{u}_{j+1}\|_{\dot{\mathbb{H}}^{1}})\Big{)}\|\widehat{u}_{j+1}-\widehat{u}_{j}\|_{L^{2}}^{2\gamma_{1}}$
$\displaystyle\times(1+\|\widehat{u}_{j+1}-\widehat{u}_{j}\|_{L^{2}}^{2-2\gamma_{1}}).$

Taking the $p$th moment for $p\geq 1$, using Lemma 4.1, Corollary 3.1 and the
definition of $\widehat{u}_{j}$, we conclude that

$\displaystyle\|\sqrt{F(\widehat{u}_{j+1})+\delta_{0}}-q_{j+1}\|_{L^{p}(\Omega)}$
$\displaystyle\leq\|q_{0}-\sqrt{F(\widehat{u}_{0})+\delta_{0}}\|_{L^{p}(\Omega)}+C\sum_{k=0}^{j}\|\widehat{u}_{j+1}-\widehat{u}_{j}\|_{L^{4p\gamma_{1}}(\Omega)}^{2\gamma_{1}}$
$\displaystyle\leq C\tau+C\tau^{2\gamma_{1}-1}\leq C\tau^{2\gamma_{1}-1}.$

∎

It can be seen that if Assumption 2.1 holds, one could take $\gamma_{1}=1$ in
Proposition 4.1.

###  4.2 Strong convergence analysis of stochastic SAV  schemes

Now, we are in a position to prove the strong convergence rate of the proposed
stochastic SAV  schemes under Assumption 2.1.

######  Theorem 4.1.

Let $X_{0}\in\mathbb{H}^{1}$ and Assumption 2.1 hold. Suppose that

$\displaystyle\|(-\Lambda)^{-\frac{1}{2}}\left(f(u)-f(\widetilde{u})\right)\|_{L^{2}}+\left\|(-\Lambda)^{-\frac{1}{2}}\left(g(\Theta
u)-g(\Theta\widetilde{u})\right)\right\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\leq
C\|u-\widetilde{u}\|_{L^{2}},$ (16)

where $u,\widetilde{u}\in\dot{\mathbb{H}}^{1}.$ Then for $p\geq 1$, the
numerical scheme (9) satisfies

$\displaystyle\sup\limits_{n\in\\{1,\ldots,N\\}}\mathbb{E}\left[\|X(t_{n})-X_{n}\|_{\mathbb{H}}^{2p}\right]\leq
C\tau^{2p},$ (17)

where $C:=C(p,X_{0},\mathbf{Q},T)>0$, $N\in\mathbb{N}^{+}$, $N\tau=T.$

###### Proof.

We only present the proof for the case $p=1$ since the proof for other cases
is similar. Fix $n\in\\{1,\ldots,N\\}.$ Notice that the solution of (9) can be
rewritten as

$\begin{split}X_{n}=&E(t_{n})X_{0}+\sum\limits_{j=0}^{n-1}E(t_{n-j})\begin{bmatrix}0\\\
g(\Theta u_{j})\delta W_{j}\end{bmatrix}\\\
&+\sum\limits_{j=0}^{n-1}E(t_{n-1-j})A^{-1}(E(\tau)-I)\begin{bmatrix}0\\\
-\frac{f(\widehat{u}_{j})}{\sqrt{F(\widehat{u}_{j})+\delta_{0}}}\frac{q_{j}+q_{j+1}}{2}\end{bmatrix}.\end{split}$
(18)

Recall that the mild solution of (LABEL:mod;swe) satisfies

$\begin{split}X(t_{n})=E(t_{n})X_{0}&+\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}E(t_{n}-s)[\mathbb{F}(X(s))ds+\mathbb{G}(X(s))dW(s)].\end{split}$
(19)

Let $\varepsilon_{i}=X(t_{i})-X_{i}$ for $i\in\\{0,1,\ldots,N\\}$.  Then

$\displaystyle\|\varepsilon_{n}\|_{\mathbb{H}}^{2}\leq$ $\displaystyle
C\Big{\|}\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}(E(t_{n}-s)-E(t_{n}-t_{j}))\begin{bmatrix}0\\\
g(\Theta u(s))dW(s)\end{bmatrix}\Big{\|}_{\mathbb{H}}^{2}$
$\displaystyle+C\Big{\|}\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}E(t_{n}-t_{j})\begin{bmatrix}0\\\
(g(\Theta u(s))-g(\Theta u_{j}))dW(s)\end{bmatrix}\Big{\|}_{\mathbb{H}}^{2}$
$\displaystyle+C\Big{\|}\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\Big{(}E(t_{n}-s)\begin{bmatrix}0\\\
-\frac{f(u(s))q(s)}{\sqrt{F(u(s))+\delta_{0}}}+\frac{f(\widehat{u}_{j})}{\sqrt{F(\widehat{u}_{j})+\delta_{0}}}\frac{q_{j}+q_{j+1}}{2}\end{bmatrix}\Big{)}ds\Big{\|}_{\mathbb{H}}^{2}$
$\displaystyle=:$ $\displaystyle Err_{n,1}+Err_{n,2}+Err_{n,3}.$

Then by Assumption 2.1, Lemmas 2.1 and 2.2, we obtain

$\displaystyle\mathbb{E}[Err_{n,1}]\leq$ $\displaystyle
C\int_{0}^{t_{n}}\mathbb{E}\left\|\left(E(s-\left[\frac{s}{\tau}\right]\tau)-I\right)\begin{bmatrix}0\\\
g(\Theta
u(s))\end{bmatrix}\right\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\mathbb{H}),\mathbb{H})}^{2}ds$
$\displaystyle\leq$ $\displaystyle
C\sum\limits_{j=0}^{N-1}\int_{t_{j}}^{t_{j+1}}\big{|}s-\left[\frac{s}{\tau}\right]\tau\big{|}^{2}\mathbb{E}\left[\left\|g(\Theta
u(s))\right\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}^{2}\right]ds\leq
C\tau^{2}.$

Similar arguments, together with Burkholder–Davis–Gundy’s inequality, yield
that

$\displaystyle\mathbb{E}[Err_{n,2}]\leq$ $\displaystyle
C\mathbb{E}\left\|\int_{0}^{t_{n}}E\left(t_{n}-\left[\frac{s}{\tau}\right]\tau\right)\begin{bmatrix}0\\\
(g(\Theta u(s))-g(\Theta
u([\frac{s}{\tau}]\tau))dW(s)\end{bmatrix}\right\|_{\mathbb{H}}^{2}$
$\displaystyle+C\mathbb{E}\left\|\int_{0}^{t_{n}}E\left(t_{n}-\left[\frac{s}{\tau}\right]\tau\right)\begin{bmatrix}0\\\
(g(\Theta u([\frac{s}{\tau}]\tau))-g(\Theta
u_{[\frac{s}{\tau}]})dW(s)\end{bmatrix}\right\|_{\mathbb{H}}^{2}$
$\displaystyle\leq$ $\displaystyle
C\int_{0}^{t_{n}}\mathbb{E}\left\|E(t_{n}-\left[\frac{s}{\tau}\right]\tau)\begin{bmatrix}0\\\
g(\Theta u(s))-g(\Theta
u(\left[\frac{s}{\tau}\right]\tau)\end{bmatrix}\right\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\mathbb{H}),\mathbb{H})}^{2}ds$
$\displaystyle+C\int_{0}^{t_{n}}\mathbb{E}\left\|E(t_{n}-\left[\frac{s}{\tau}\right]\tau)\begin{bmatrix}0\\\
g(\Theta u([\frac{s}{\tau}]\tau))-g(\Theta
u_{[\frac{s}{\tau}]})\end{bmatrix}\right\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\mathbb{H}),\mathbb{H})}^{2}ds.$

Thanks to (16) and Proposition 2.2, we have

$\displaystyle\mathbb{E}[Err_{n,2}]\leq$ $\displaystyle
C\int_{0}^{t_{n}}\mathbb{E}\left\|(-\Lambda)^{-\frac{1}{2}}\left(g(u(s))-g\left(u\left(\left[\frac{s}{\tau}\right]\tau\right)\right)\right)\right\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}^{2}ds$
$\displaystyle+C\int_{0}^{t_{n}}\mathbb{E}\left\|(-\Lambda)^{-\frac{1}{2}}\left(g\left(u\left(\left[\frac{s}{\tau}\right]\tau\right)\right)-g(u_{\left[\frac{s}{\tau}\right]})\right)\right\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}^{2}ds$
$\displaystyle\leq$ $\displaystyle
C\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\mathbb{E}\|u(s)-u(t_{j})\|_{L^{2}}^{2}ds+C\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\mathbb{E}\|u(t_{j})-u_{j}\|_{L^{2}}^{2}ds$
$\displaystyle\leq$ $\displaystyle
C\tau^{2}+C\tau\sum\limits_{j=0}^{n-1}\mathbb{E}\left\|u(t_{j})-u_{j}\right\|_{L^{2}}^{2}.$

Now we turn to consider $Err_{n,3}.$ Based on Lemma 2.3, we obtain

$\displaystyle\mathbb{E}[Err_{n,3}]$ (20) $\displaystyle\leq$ $\displaystyle
C\mathbb{E}\left[\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\left\|f(u(s))-\frac{f(\widehat{u}_{j})q_{j}}{\sqrt{F(\widehat{u}_{j})+\delta_{0}}}\right\|_{\dot{\mathbb{H}}^{-1}}^{2}ds\right]+C\mathbb{E}\left[\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\left\|\frac{f(\widehat{u}_{j})(q_{j+1}-q_{j})}{2\sqrt{F(\widehat{u}_{j})+\delta_{0}}}\right\|_{\dot{\mathbb{H}}^{-1}}^{2}ds\right]$
$\displaystyle\leq$ $\displaystyle
C\mathbb{E}\left[\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\left\|f(u(s))-f(u(t_{j}))\right\|_{\dot{\mathbb{H}}^{-1}}^{2}ds\right]+C\mathbb{E}\left[\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\left\|f(u(t_{j}))-f(\widehat{u}_{j})\right\|_{\dot{\mathbb{H}}^{-1}}^{2}ds\right]$
$\displaystyle+C\mathbb{E}\left[\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\left\|f(\widehat{u}_{j})-\frac{f(\widehat{u}_{j})q_{j}}{\sqrt{F(\widehat{u}_{j})+\delta_{0}}}\right\|_{\dot{\mathbb{H}}^{-1}}^{2}ds\right]+C\mathbb{E}\left[\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\left\|\frac{f(\widehat{u}_{j})(q_{j+1}-q_{j})}{\sqrt{F(\widehat{u}_{j})+\delta_{0}}}\right\|_{\dot{\mathbb{H}}^{-1}}^{2}ds\right]$
$\displaystyle=:$ $\displaystyle{\rm II}_{n}^{1}+{\rm II}_{n}^{2}+{\rm
II}_{n}^{3}+{\rm II}_{n}^{4}.$

According to (16) and Proposition 2.2, we derive

$\displaystyle{\rm II}_{n}^{1}\leq$ $\displaystyle
C\mathbb{E}\left[\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\|u(s)-u(t_{j})\|_{L^{2}}^{2}ds\right]\leq
C\tau^{2},$ $\displaystyle{\rm II}_{n}^{2}\leq$ $\displaystyle
C\tau\sum\limits_{j=0}^{n-1}\mathbb{E}\left[\|u(t_{j})-u_{j}\|_{L^{2}}^{2}\right]+C\tau\sum\limits_{j=0}^{n-1}\mathbb{E}\left[\|u_{j}-\widehat{u}_{j}\|_{L^{2}}^{2}\right].$

If $\widehat{u}_{j}=u_{j},$ then ${\rm II}_{n}^{2}\leq
C\tau\sum\limits_{j=0}^{n-1}\mathbb{E}\left[\|u(t_{j})-u_{j}\|_{L^{2}}^{2}\right].$
When $\widehat{u}_{j}=\frac{3u_{j}-u_{j-1}}{2},$ thanks to $u_{0}=u_{-1},$ we
have

$\displaystyle{\rm II}_{n}^{2}\leq$ $\displaystyle
C\tau\sum\limits_{j=0}^{n-1}\mathbb{E}\left[\|u(t_{j})-u_{j}\|_{L^{2}}^{2}\right]+C\tau\sum\limits_{j=0}^{n-1}\mathbb{E}\left[\|u_{j}-u_{j-1}\|_{L^{2}}^{2}\right]$
$\displaystyle\leq$ $\displaystyle
C\tau\sum\limits_{j=0}^{n-1}\mathbb{E}\left[\|u(t_{j})-u_{j}\|_{L^{2}}^{2}\right]+C\tau\sum\limits_{j=0}^{n-2}\mathbb{E}\left[\|u_{j+1}-u_{j}\|_{L^{2}}^{2}\right].$

For the term ${\rm II}_{n}^{4},$ we obtain

$\displaystyle{\rm II}_{n}^{4}\leq$ $\displaystyle
C\mathbb{E}\left[\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\frac{\|f(\widehat{u}_{j})\|_{\dot{\mathbb{H}}^{-1}}^{2}}{F(\widehat{u}_{j})+\delta_{0}}|q_{j+1}-q_{j}|^{2}ds\right]$
$\displaystyle\leq$ $\displaystyle
C\mathbb{E}\left[\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\frac{\|f(\widehat{u}_{j})\|_{\dot{\mathbb{H}}^{-1}}^{2}}{(F(\widehat{u}_{j})+\delta_{0})^{2}}\|f(\widehat{u}_{j})\|_{L^{2}}^{2}\|u_{j+1}-u_{j}\|_{L^{2}}^{2}ds\right].$

Since $\|f(\widehat{u}_{j})\|\leq
b_{1}(\|\widehat{u}_{j}\|_{\dot{\mathbb{H}}^{1}})$ and
$F(\widehat{u}_{j})+\delta_{0}\geq c>0,$ the Hölder’s inequality, Lemma 4.1
and Corollary 3.1 yield that

$\displaystyle{\rm II}_{n}^{4}\leq
C\tau\sum\limits_{j=0}^{n-1}\sqrt{\mathbb{E}\left[\|u_{j+1}-u_{j}\|_{L^{2}}^{4}\right]}\leq
C\tau^{2}.$

By applying Proposition 4.1, Corollary 3.1 and Hölder’s inequality, we get

$\displaystyle{\rm II}_{n}^{3}\leq$ $\displaystyle
C\mathbb{E}\left[\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\left\|f(\widehat{u}_{j})-\frac{f(\widehat{u}_{j})q_{j}}{\sqrt{F(\widehat{u}_{j})+\delta_{0}}}\right\|_{\dot{\mathbb{H}}^{-1}}^{2}ds\right]$
$\displaystyle\leq$ $\displaystyle
C\mathbb{E}\left[\sum\limits_{j=0}^{n-1}\int_{t_{j}}^{t_{j+1}}\frac{\|f(\widehat{u}_{j})\|_{\dot{\mathbb{H}}^{-1}}^{2}}{F(\widehat{u}_{j})+\delta_{0}}|\sqrt{F(\widehat{u}_{j})+\delta_{0}}-q_{j}|^{2}ds\right]$
$\displaystyle\leq$ $\displaystyle C\tau^{2}.$

Combining the above estimates together, we arrive at

$\displaystyle\mathbb{\|}\varepsilon_{n}\|_{\mathbb{H}}^{2}\leq$
$\displaystyle
C\tau^{2}+C\tau\sum\limits_{j=0}^{n-1}\mathbb{E}\left\|u(t_{j})-u_{j}\right\|_{L^{2}}^{2}.$

By discrete Grönwall’s inequality and Lemma 4.1, we complete the proof. ∎

Next, we present the strong convergence rate of (8), which can be shown by
using the similar steps as in Theorem 4.1, [13, Lemmas 4.1 and 4.2] and
Proposition 4.1. We omit its proof for convenience.

######  Theorem 4.2.

Let $X_{0}\in\mathbb{H}^{\gamma},\gamma\in[1,2],$ and Assumption 2.1 hold.
Suppose that

$\displaystyle\|(-\Lambda)^{-\frac{1}{2}}\left(f(u)-f(\widetilde{u})\right)\|_{L^{2}}+\left\|(-\Lambda)^{-\frac{1}{2}}\left(g(\Theta
u)-g(\Theta\widetilde{u})\right)\right\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\leq
C\|u-\widetilde{u}\|_{L^{2}},$
$\displaystyle\|(-\Lambda)^{\frac{\gamma-1}{2}}f(u)\|_{L^{2}}+\|(-\Lambda)^{\frac{\gamma-1}{2}}g(u)\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\leq
b_{6}(\|u\|_{\dot{\mathbb{H}}^{1}}),$ (21)

where $b_{6}$ is a polynomial, $u,\widetilde{u}\in\dot{\mathbb{H}}^{1}.$ Then
for $p\geq 1$, the numerical scheme (8) satisfies

$\displaystyle\sup\limits_{n\in\\{1,\ldots,N\\}}\mathbb{E}\left[\|X(t_{n})-X_{n}\|_{\mathbb{H}}^{2p}\right]\leq
C\tau^{p\gamma},$ (22)

where $C:=C(p,X_{0},\mathbf{Q},T)>0$, $N\in\mathbb{N}^{+}$, $N\tau=T.$

######  Remark 4.1.

Schemes (8) and (9) can also be used to numerically solve the deterministic
wave equation, which corresponds to the case of $g=0$. In this case, the above
convergence result of SAV schemes is also new.

##  5 Semi-implicit energy-preserving fully-discrete schemes

In this section, we combine the stochastic SAV  schemes with a spatial finite
element method (see, e.g., [1, 22]) to propose implementary and energy-
preserving fully-discrete schemes for (2). We first briefly recall the
definition of the linear finite element method. Let
$\left\\{\mathcal{T}_{h}\right\\}$ be a quasi-uniform family of triangulations
of the convex polygonal domain $\mathcal{O}$ with
$h_{\mathrm{K}}=\operatorname{diam}(K)$ and
$h=\max\limits_{\mathrm{K}\in\mathcal{T}_{h}}h_{\mathrm{K}}.$ Let
$V_{h}\subset\dot{\mathbb{H}}^{1}$ be the space of piecewise linear continuous
functions with respect to $\mathcal{T}_{h}$ which are zero on the boundary of
$\mathcal{D}$, and let $\mathcal{P}_{h}:{\dot{\mathbb{H}}}\rightarrow V_{h}$
denote the $\dot{\mathbb{H}}$-orthogonal projector and
$\mathcal{R}_{h}:\dot{\mathbb{H}}^{1}\rightarrow V_{h}$ denote the
$\dot{\mathbb{H}}^{1}$-orthogonal projector (Ritz projector). Thus,

$\displaystyle\left(\mathcal{P}_{h}v,w_{h}\right)=\left(v,w_{h}\right),\quad\left(\nabla\mathcal{R}_{h}u,\nabla
w_{h}\right)=\left(\nabla u,\nabla
w_{h}\right)\quad\forall\;v\in\dot{\mathbb{H}},u\in\dot{\mathbb{H}}^{1},w_{h}\in
V_{h}.$

The discrete Laplace operator $\Lambda_{h}:V_{h}\rightarrow V_{h}$ is then
defined by

$\displaystyle\left(\Lambda_{h}v_{h},w_{h}\right)=-\left(\nabla v_{h},\nabla
w_{h}\right)\quad\forall\;w_{h}\in V_{h}.$

Notice that $\mathcal{R}_{h}=(-\Lambda_{h})^{-1}\mathcal{P}_{h}(-\Lambda)$
(see, e.g., [22]). We define discrete norms and interpolation spaces by

$\displaystyle\left\|v_{h}\right\|_{h,\alpha}=\left\|(-\Lambda_{h})^{\frac{\alpha}{2}}v_{h}\right\|\quad\forall\;v_{h}\in
V_{h}$

and $\dot{\mathbb{H}}_{h}^{\alpha}=V_{h}$ equipped with the norm
$\|\cdot\|_{h,\alpha}$, respectively. Then the finite element method of (2)
becomes

$\left\\{\begin{aligned}
&dX^{h}(t)=A_{h}X^{h}(t)dt+\mathcal{P}_{h}\mathbb{F}\left(X^{h}(t)\right)dt+\mathcal{P}_{h}\mathbb{G}\left(X^{h}(t)\right)dW(t),\quad
t>0,\\\ &X^{h}(0)=X^{h,0},\end{aligned}\right.$ (23)

where $X^{h,0}=(u^{h,0},v^{h,0})^{\top},$
$u^{h,0}=\mathcal{R}_{h}u_{0},v^{h,0}=\mathcal{P}_{h}v_{0}$,

$\displaystyle A_{h}:=\begin{bmatrix}0&I\\\
\Lambda_{h}&0\end{bmatrix}\quad\text{and}\quad X^{h}:=\begin{bmatrix}u_{h}\\\
v_{h}\end{bmatrix}.$

Here the notation
$\mathcal{P}_{h}\mathbb{F}(X^{h})=(0,\mathcal{P}_{h}f(u^{h}))^{\top}$ and
similarly for $\mathcal{P}_{h}\mathbb{G}(X^{h}).$ Like (3), the mild form of
$X^{h}$ reads

$\displaystyle
X^{h}(t)=E_{h}(t)X^{h,0}+\int_{0}^{t}E_{h}(t-s)\mathcal{P}^{h}\mathbb{F}(X^{h}(s))ds+\int_{0}^{t}E_{h}(t-s)\mathcal{P}^{h}\mathbb{G}(X^{h}(s))dW(s),$

where $E_{h}=\begin{bmatrix}C_{h}(t)&(-\Lambda_{h})^{-\frac{1}{2}}S_{h}(t)\\\
-(-\Lambda_{h})^{\frac{1}{2}}S_{h}(t)&C_{h}(t)\end{bmatrix}$ is a
$\mathbb{C}_{0}$-semigroup generated by $A_{h}$ on
$\mathbb{H}_{h}:=\dot{\mathbb{H}}^{0}_{h}\times\dot{\mathbb{H}}^{-1}_{h}$ with
$C_{h}(t)=\cos(t(-\Lambda_{h})^{\frac{1}{2}})$ and
$S_{h}(t)=\sin(t(-\Lambda_{h})^{\frac{1}{2}}).$ For (23), the corresponding
energy is defined by

$\displaystyle
H(X^{h})=\frac{1}{2}\left\|(-\Lambda_{h})^{\frac{1}{2}}u^{h}\right\|^{2}+\frac{1}{2}\left\|v^{h}\right\|^{2}+F(u^{h}),$

since $\left\|\nabla
u^{h}\right\|^{2}=\left\|(-\Lambda_{h})^{\frac{1}{2}}u^{h}\right\|^{2}$. By
using the Itô formula, one can obtain the following averaged energy evolution
of the finite element solution $X^{h}$. Similar to the continuous case, the
regularity estimate of the mild solution follows.

######  Proposition 5.1.

Let Assumption 2.2 hold and $X_{0}\in\mathbb{H}^{1}$. The solution $X^{h}$ of
the finite element approximation (23) satisfies the averaged energy evolution
law,

$\displaystyle\mathbb{E}\left[H(X^{h}(t))\right]=$
$\displaystyle\mathbb{E}\left[H(X^{h}(0))\right]$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\mathbb{E}\Big{[}\operatorname{Tr}\left(\mathcal{P}_{h}g(\Theta
u^{h}(s){\bf Q}^{\frac{1}{2}})(\mathcal{P}_{h}g(\Theta u^{h}(s)){\bf
Q}^{\frac{1}{2}})^{*}\right)\Big{]}ds,\quad t\geq 0.$

Furthermore, suppose that $X_{0}\in\mathbb{H}^{\beta}$ for some
$\beta\in[1,2]$ and that

$\displaystyle\|(-\Lambda)^{\frac{\beta-1}{2}}f(u)\|_{L^{2}}+\|(-\Lambda)^{\frac{\beta-1}{2}}g(\Theta
u)\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\leq
b_{4}(\|u\|_{\dot{\mathbb{H}}^{1}}),\;u\in\dot{\mathbb{H}}^{\beta},$ (24)

where $b_{4}$ is a polynomial. Then it holds that for any $p\geq 2,$

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}\Big{[}\|X^{h}(t)\|_{\mathbb{H}_{h}^{\beta}}^{p}\Big{]}\leq
C(p,X_{0},\mathbf{Q},T),$

and for $0\leq s\leq t\leq T,$

$\displaystyle\mathbb{E}\Big{[}\|u^{h}(t)-u^{h}(s)\|_{\dot{\mathbb{H}}_{h}^{\small
0}}^{p}\Big{]}\leq C(p,X_{0},\mathbf{Q},T)|t-s|^{p},$
$\displaystyle\mathbb{E}\Big{[}\|v^{h}(t)-v^{h}(s)\|_{\dot{\mathbb{H}}_{h}^{\small-1}}^{p}\Big{]}\leq
C(p,X_{0},\mathbf{Q},T)|t-s|^{\frac{p}{2}},$

where $C(p,X_{0},\mathbf{Q},T)$ is a positive constant depending on
$p,X_{0},\mathbf{Q},$ and $T$.

Next, we apply the SAV  scheme (9) to (23) and obtain the following fully-
discrete scheme

$\displaystyle
X^{h}_{n+1}=E_{h}(\tau)X^{h}_{n}+A_{h}^{-1}(E_{h}(\tau)-I)\mathcal{P}_{h}\Big{(}0,-\frac{f(\widehat{u}^{h}_{n})}{\sqrt{F(\widehat{u}^{h}_{n})+\delta_{0}}}\frac{q_{n}+q_{n+1}}{2}\Big{)}^{\top}$
(25) $\displaystyle\quad\quad\quad+E_{h}(\tau)\mathcal{P}_{h}(0,g(\Theta
u^{h}_{n})\delta W_{n})^{\top},$ $\displaystyle
q_{n+1}=q_{n}+\frac{(f(\widehat{u}_{n}^{h}),u_{n+1}^{h}-u_{n}^{h})}{2\sqrt{F(\widehat{u}_{n}^{h})+\delta_{0}}}.$

Here $\widehat{u}_{n}^{h}=u_{n}^{h}$ (or $\frac{3u_{n}^{h}-u_{n-1}^{h}}{2}$,
with $u_{-1}^{h}=u_{0}^{h}$), $T>0$, $N\in\mathbb{N}^{+}$, $N\tau=T,n\leq N$.
Following the procedures in the proof of Theorem 4.1, we could get the
following strong convergence rate result of (25). Below we only present a
sketch proof due to the limitation of pages.

######  Theorem 5.1.

Under Assumption 2.1, let $T>0$, $N\in\mathbb{N}^{+}$, $N\tau=T,n\leq N$.
Assume that $f,g$ satisfy (24) and (16) with some $\beta\geq 1$, and that
$X_{0}\in\mathbb{H}^{\gamma},$ $\gamma\geq 1+\frac{2}{3}\beta.$ Then the
numerical scheme (25) satisfies that for $p\geq 2,$

$\displaystyle\mathbb{E}\Big{[}\|X^{h}_{n}-X(t_{n})\|_{\mathbb{H}_{h}}^{p}\Big{]}\leq
C(p,X_{0},\mathbf{Q},T)(h^{\frac{2}{3}\beta p}+\tau^{p}).$

###### Proof.

We decompose $X^{h}_{n}-X(t_{n})$ by

$X^{h}_{n}-X(t_{n})=X^{h}_{n}-X^{h}(t_{n})+X^{h}(t_{n})-X(t_{n}).$

The estimate of
$\mathbb{E}\Big{[}\|X^{h}(t_{n})-X(t_{n})\|_{\mathbb{H}_{h}}^{p}\Big{]}\leq
C(p,X_{0},\mathbf{Q},T)h^{\frac{2}{3}\beta p}$ could be established by using
similar arguments as in the proof of [1, Theorem 3.1]. For the term
$X^{h}_{n}-X^{h}(t_{n})$, one may follow same steps as in proving Theorem 4.1,
together with the properties of the discrete cosine and sine operators (see,
e.g., [1]), and obtain

$\mathbb{E}\Big{[}\|X^{h}_{n}-X^{h}(t_{n})\|_{\mathbb{H}_{h}}^{p}\Big{]}\leq
C(p,X_{0},\mathbf{Q},T)\tau^{p}.$

Combining the above estimates, we complete the proof. ∎

Following the above approach, one could obtain the analogous result for the
fully-discrete scheme based on (8) and linear finite element method, i.e.,

$\displaystyle
u_{n+1}^{h}=u_{n}^{h}+\frac{h}{2}(v_{n}^{h}+v_{n+1}^{h})+\frac{h}{2}\mathcal{P}_{h}g(\Theta
u^{h}_{n})\delta W_{n},$ (26) $\displaystyle
v_{n+1}^{h}=v_{n}^{h}+\frac{h}{2}\Lambda_{h}(u_{n}^{h}+u_{n+1}^{h})-h\frac{\mathcal{P}_{h}f(\widehat{u}_{n}^{h})}{\sqrt{F(\widehat{u}_{n}^{h})+\delta_{0}}}\frac{q_{n}+q_{n+1}}{2}+\mathcal{P}^{h}g(\Theta
u_{n}^{h})\delta W_{n},$ $\displaystyle
q_{n+1}=q_{n}+\frac{(f(\widehat{u}_{n}^{h}),u^{h}_{n+1}-u^{h}_{n})}{2\sqrt{F(\widehat{u}_{n}^{h})+\delta_{0}}},$

where $\widehat{u}_{n}^{h}=u_{n}^{h}$ (or $\frac{3u_{n}^{h}-u_{n-1}^{h}}{2}$
with $u_{-1}^{h}=u_{0}^{h}$), $T>0$, $N\in\mathbb{N}^{+}$, $N\tau=T,n\leq N$.
In the end of this section, we present the discrete energy evolution of (25)
and its weak error estimate.

######  Proposition 5.2.

Let the condition of Theorem 5.1 hold and $X_{0}\in\mathbb{H}^{1}$, $T>0$,
$N\in\mathbb{N}^{+}$, $N\tau=T,n\leq N.$ The proposed  scheme (25) preserves
the discrete averaged modified energy evolution law, i.e., for
$n\in\\{0,1,\ldots,N-1\\},$

$\displaystyle\mathbb{E}[\widetilde{H}(u_{n+1}^{h},v_{n+1}^{h},q_{n+1})]=$
$\displaystyle\mathbb{E}[\widetilde{H}(u_{n}^{h},v_{n}^{h},q_{n})]+\frac{\tau}{2}\mathbb{E}\Big{[}{\rm{Tr}}\left(\mathcal{P}_{h}g(\Theta
u_{n}^{h})\mathbf{Q}^{\frac{1}{2}}(\mathcal{P}_{h}g(\Theta
u_{n}^{h})\mathbf{Q}^{\frac{1}{2}})^{*}\right)\Big{]}.$

Here
$\widetilde{H}(u_{n}^{h},v_{n}^{h},q_{n})=\frac{1}{2}\|(-\Lambda_{h})^{\frac{1}{2}}u_{n}^{h}\|^{2}+\frac{1}{2}\|v_{n}^{h}\|^{2}+q_{n}^{2}.$
In addition, assume that

$\displaystyle\left\|\left(g(\Theta
u)-g(\Theta\widetilde{u})\right)\right\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\leq
C\|u-\widetilde{u}\|,$ (27)
$\displaystyle\|(-\Lambda)^{\frac{\beta_{1}}{2}}g(\Theta
u)\|_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\leq
Cb_{7}(\|u\|_{\dot{\mathbb{H}}^{\beta_{1}}}),\quad\beta_{1}\in[1,2].$

Then (25) satisfies

$\displaystyle\Big{|}\mathbb{E}\Big{[}\widetilde{H}(u_{n}^{h},v_{n}^{h},q_{n})-V_{1}(u(t_{n}),v(t_{n}))\Big{]}\Big{|}\leq
C(p,T,X_{0})(|err_{0}|+h^{\beta_{1}}+h^{\frac{2}{3}\beta}+\tau),$

where  $\beta\in[1,3],$
$err_{0}=\big{|}\widetilde{H}(u_{0}^{h},v_{0}^{h},q_{0})-V_{1}(u(0),v(0))\big{|}.$

###### Proof.

Following similar arguments as in the proof of Proposition 3.1, one can get
the discrete energy evolution of (25),

$\displaystyle\mathbb{E}[\widetilde{H}(u_{n+1}^{h},v_{n+1}^{h},q_{n+1})]=$
$\displaystyle\mathbb{E}[\widetilde{H}(u_{n}^{h},v_{n}^{h},q_{n})]+\frac{\tau}{2}\mathbb{E}\Big{[}{\rm{Tr}}\left(\mathcal{P}^{h}g(\Theta
u^{h}_{n})\mathbf{Q}^{\frac{1}{2}}(\mathcal{P}_{h}g(\Theta
u_{n})\mathbf{Q}^{\frac{1}{2}})^{*}\right)\Big{]},$

which implies the first assertion. According to (5), we have

$\displaystyle\mathbb{E}\Big{[}\widetilde{H}(u_{n}^{h},v_{n}^{h},q_{n})-V_{1}(u(t_{n}),v(t_{n}))\Big{]}$
$\displaystyle=$ $\displaystyle
err_{0}+\frac{1}{2}\int_{0}^{t_{n}}\mathbb{E}\Big{[}{\rm{Tr}}\left(\mathcal{P}_{h}g(\Theta{u_{[\frac{s}{\tau}]}^{h}})\mathbf{Q}^{\frac{1}{2}}(\mathcal{P}_{h}g(\Theta{u_{[\frac{s}{\tau}]}^{h}})\mathbf{Q}^{\frac{1}{2}})^{*}\right)$
$\displaystyle-{\rm{Tr}}\left(g(\Theta u(s))\mathbf{Q}^{\frac{1}{2}}(g(\Theta
u(s))\mathbf{Q}^{\frac{1}{2}})^{*}\right)\Big{]}ds.$

From (27), Proposition 2.2, Corollary 3.1, Hölder’s inequality and
$\|(I-P_{h})w\|\leq Ch^{\beta}\|w\|_{\dot{\mathbb{H}}^{\beta}},\beta\in[1,2]$,
it follows that

$\displaystyle\quad\Big{|}\mathbb{E}\Big{[}\widetilde{H}(u_{n}^{h},v_{n}^{h},q_{n})-V_{1}(u(t_{n}),v(t_{n}))\Big{]}\Big{|}$
$\displaystyle\leq|err_{0}|+C{\int_{0}^{t_{n}}}\Big{|}\mathbb{E}\Big{[}{\rm{Tr}}\left((I-\mathcal{P}^{h})g(\Theta
u(s))\mathbf{Q}^{\frac{1}{2}}((I+\mathcal{P}_{h})g(\Theta
u(s))\mathbf{Q}^{\frac{1}{2}})^{*}\right)\Big{]}\Big{|}ds$
$\displaystyle+C{\int_{0}^{t_{n}}}\Big{|}\mathbb{E}\Big{[}{\rm{Tr}}\left(\mathcal{P}^{h}\big{(}g(\Theta
u(s))-g(\Theta{u_{[\frac{s}{\tau}]}^{h}})\big{)}\mathbf{Q}^{\frac{1}{2}}(\mathcal{P}_{h}\big{(}g(\Theta
u(s))+g({u_{[\frac{s}{\tau}]}^{h}})\big{)}\mathbf{Q}^{\frac{1}{2}})^{*}\right)\Big{]}\Big{|}ds$
$\displaystyle\leq|err_{0}|+C{\int_{0}^{t_{n}}}\mathbb{E}\Big{[}\Big{\|}(I-\mathcal{P}_{h})g(\Theta
u(s))\Big{\|}_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\Big{\|}g(\Theta
u(s))\Big{\|}_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\Big{]}ds$
$\displaystyle+C{\int_{0}^{t_{n}}}\mathbb{E}\Big{[}\Big{\|}g(\Theta
u(s))-g(\Theta{u_{[\frac{s}{\tau}]}^{h}})\Big{\|}_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\Big{\|}g(\Theta
u(s))+g(\Theta{u_{[\frac{s}{\tau}]}^{h}})\Big{\|}_{\mathcal{L}_{2}(\mathbf{Q}^{\frac{1}{2}}(\dot{\mathbb{H}}),\dot{\mathbb{H}})}\Big{]}ds$
$\displaystyle\leq|err_{0}|+Ch^{\beta_{1}}+\int_{0}^{t_{n}}\sqrt{\mathbb{E}\Big{[}\|{u_{[\frac{s}{\tau}]}^{h}}-u(s)\|^{2}\Big{]}}ds.$

Applying the temporal regularity estimate in Proposition 2.2 and Theorem 5.1,
we conclude that

$\displaystyle\Big{|}\mathbb{E}\Big{[}\widetilde{H}(u_{n}^{h},v_{n}^{h},q_{n})-V_{1}(u(t_{n}),v(t_{n}))\Big{]}\Big{|}$
$\displaystyle\leq|err_{0}|+Ch^{\beta_{1}}+C(\tau+h^{\frac{2}{3}\beta}).$

∎

It can be seen that the weak error of the energy for (25) is only determined
by the spatial discretization in the additive noise case since there is no
weak error in the temporal direction in this case (see Proposition 3.1).

##  6 Numerical experiments

This section presents numerical experiments to illustrate the strong
convergence order and  the preservation of energy evolution law of the
proposed numerical schemes (25) and (26) with $\hat{u}_{n}=u_{n},$
$n\in\\{0,1,\ldots,N\\},$ for 1-dimensional nonlinear SWE under the
homogeneous Dirichlet boundary condition,

$\left\\{\begin{aligned} &du=vdt,&&(x,t)\in(0,1)\times(0,T],\\\
&dv=u_{xx}dt-f(u)dt+g(u)dW(t),&&(x,t)\in(0,1)\times(0,T],\\\ &u(x,0)=\sin(\pi
x),\quad v(x,0)=0.\end{aligned}\right.$ (28)

Figure 1: The strong convergence order in temporal direction $($left$)$
$f(u)=u,g(u)=\sin(u)$, $($right$)$ $f(u)=\sin(u),g(u)=\sin(u)$

Fig. 1 displays the temporal approximation errors $\|u(T)-U^{h}_{N}\|_{L^{2}}$
against $\tau$ on log-log scale with $\tau=2^{-\bf s},\,{\bf s}=8,9,10,11,12$
at time $T=1$ for multiplicative noise, respectively. We fix the spatial step
size $h=2^{-6}$ and simulate the exact solution with the numerical one by
using a small step size $\tau=2^{-14}.$ It can be observed that the slopes of
numerical schemes are closed to 1, which implies that the temporal convergence
order of the proposed numerical schemes (25) and (26) is 1. Here the
expectation is approximated by taking average over 1000 realizations.

Figure 2: Averaged energy evolution relationship $($left$)$ $f(u)=u,$
$($right$)$ $f(u)=\sin(u)$

Fig. 2 presents the evolution of discrete averaged energies for the proposed
numerical schemes (25) and (26). Here, the expectation is approximated by
taking average over 5000 realizations. From Fig. 2, it can be seen that
averaged energies associated with numerical solutions grow linearly with the
time raising when the SWE is driven by additive noise. The numerical results
are consistent with the theoretical results.

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