Gunnar Birke, Christian Engwer, Sandra May 11institutetext: Münster
University, Applied Mathematics, 11email<EMAIL_ADDRESS>22institutetext: Uppsala University, Department of Information Technology
33institutetext: Dortmund University, Fakultät für Mathematik

# Domain of Dependence stabilization for the acoustic wave equation on 2D cut-
cell meshes

Gunnar Birke 11 Christian Engwer 11 Sandra May 22 Florian Streitbürger 33

###### Abstract

Cut-cell meshes are an attractive alternative to avoid common mesh generation
problems. For hyperbolic problems they pose additional challenges, as elements
can become arbitrarily small, leading to prohibitive time step restrictions
for explicit time stepping methods. To alleviate this small cell problem we
consider a particular stabilization method, the Domain of Dependence (DoD)
method. So far, while posessing many favorable theoretical properties, in two
dimensions the DoD method was essentially restricted to the transport
equation. In this work we extend the DoD method to the acoustic wave equation
in two dimensions and provide numerical results for validation.

###### keywords:

cut-cells, small cell problem, discontinuous Galerkin method, DoD
stabilization, wave equation

##  1 Introduction

The generation of traditional body fitted meshes can become very involved and
time consuming, when working with complex geometries. One possible alternative
is the use of cut-cell meshes. The drawback is that one has no control over
the cut-cell shapes, in particular cut-cells can become arbitrarily small.
Explicit time stepping methods, which are commonly used to simulate hyperbolic
conservation laws, require a time step size that is chosen based on the
smallest cell in the mesh. This becomes infeasible on a cut-cell mesh.
Instead, one likes to choose the time step based on the size of the larger
uncut cells. This is referred to as the small cell problem.

As discontinuous Galerkin (dG) methods are attractive to solve hyperbolic
partial differential equations (PDEs), different stabilization approaches have
been developed in recent years to handle the small cell problem on cut-cell
meshes, see for example [8], [11]. We consider the Domain of Dependence (DoD)
stabilization method which was introduced in [6] for the linear transport
equation and extended, in one space dimension, to non-linear equations in
[10]. In two dimensions, the method was originally restricted to certain
flow/geometry combinations, e. g. a flow parallel to a domain boundary. In [4]
the stabilization was generalized for $P^{0}$ discretizations of linear
systems to handle triangular cut-cells with multiple inflow or outflow faces.

In this contribution we extend the lowest order DoD stabilization such that it
can be applied to the acoustic wave equation. The main difference is that the
wave equation cannot be written as a system of coupled transport equations, as
the system is not globally diagonalizable. This adds significant complications
in deriving the stabilization term. We focus on a $P^{0}$ dG discretization
here, the development of appropriate higher-order stabilization terms for the
acoustic wave equation is ongoing research. The construction of the DoD
stabilization went hand in hand with an $L^{2}$-stability analysis for the
semi-discretization in space, see [10, 12, 4], and the terms were designed
such that we regain the spatial stability properties of the original dG
scheme. This also holds true for the acoustic wave equation and the analysis
was an important cornerstone in the development of the stabilization term. As
the actual $L^{2}$-stability proof goes beyond the scope of this paper, we add
appropriate remarks where necessary to understand the design of the actual
formulation.

The outline of the paper is as follows: We first describe the problem setup
and the underlying dG scheme. Then we introduce the DoD stabilization for the
acoustic wave equation. We conclude with numerical results to validate our
findings.

##  2 Problem setup

Let $\Omega\subset\mathbb{R}^{2}$ be an open domain. We consider the acoustic
wave equation for the solution $u=(p,v_{1},v_{2})$ given by

$\displaystyle u_{t}+A_{1}u_{x}+A_{2}u_{y}$ $\displaystyle=0\>\>\>\text{ in
}\Omega,$ $\displaystyle\tau u$ $\displaystyle=g\>\>\text{ on
}\partial\Omega,$ $\displaystyle u(\cdot,0)$ $\displaystyle=u_{0}\>\>\text{ in
}\Omega,$

with $p$ being the pressure, $v=(v_{1},v_{2})^{t}$ being the velocity and
$\tau$ being a boundary operator imposing an inflow boundary condition for
incoming waves. The system matrices are given as

$A_{1}=\footnotesize\begin{pmatrix}0&c&0\\\ c&0&0\\\
0&0&0\end{pmatrix}\normalsize\text{ and
}A_{2}=\footnotesize\begin{pmatrix}0&0&c\\\ 0&0&0\\\ c&0&0\end{pmatrix}.$

Here $c>0$ denotes the speed of sound. We want to stress that $A_{1}A_{2}\neq
A_{2}A_{1}$.

In our numerical tests we choose $\Omega=[0,1]^{2}$ and discretize it by a
structured grid $\widehat{\mathcal{M}_{h}}$. We then introduce an artificial
cut, a straight line going through the square, starting at $(x_{0},0)$ and
having an angle $\gamma$ relative to the $x$-axis. This creates an internal
boundary with two subdomains which we will resolve by a cut-cell mesh
$\mathcal{M}_{h}$. A sketch is contained in Fig. 1. The wave equation is then
solved in the whole domain $\Omega$.

Our discrete function space is defined as

$\mathcal{V}_{h}=\mathcal{V}_{h}^{0}=\\{v_{h}\in
L^{2}(\Omega)^{3}:(v_{h})_{i}\big{|}_{E}\in\mathcal{P}^{0}(E)\,\forall\,1\leq
i\leq 3,\,E\in\mathcal{M}_{h}\\}.$

${\widehat{\mathcal{M}}}_{h}$$\cap$${\bar{\Omega}}$$(x_{0},0)$$\gamma$$=$${\mathcal{M}_{h}}$$E\in\widehat{\mathcal{M}}_{h}$$E_{1}$$E_{2}$
Figure 1: Construction of the mesh: Out of the structured grid
$\widehat{\mathcal{M}_{h}}$ on the domain $\Omega$ the mesh $\mathcal{M}_{h}$
is constructed by introducing cut-cells $E_{1},E_{2}\subset
E\in\widehat{M_{h}}$ along the cut such that
$\bar{E_{1}}\cup\bar{E_{2}}=\bar{E}$.

We define the sets of internal and external faces as

$\displaystyle\mathcal{F}^{\text{int}}_{h}$ $\displaystyle=\\{F=\partial
E_{1}\cap\partial E_{2}:E_{1},E_{2}\in\mathcal{M}_{h},\;E_{1}\neq
E_{2},\;|F|>0\\},$ $\displaystyle\mathcal{F}^{\text{ext}}_{h}$
$\displaystyle=\\{F=\partial
E\cap\partial\Omega:E\in\mathcal{M}_{h},\;|F|>0\\}$

and the set of neighbor cells
$\mathcal{N}(E):=\\{E^{\prime}\in\mathcal{M}_{h}:|\bar{E}^{\prime}\cap\bar{E}|>0\\}.$

So for any internal face $F\in\mathcal{F}^{\text{int}}_{h}$ there are always
two unique elements $E_{1},E_{2}\in\mathcal{M}_{h}$ such that
$\bar{E}_{1}\cap\bar{E}_{2}=F$. This face will often be denoted by
$F_{E_{1},E_{2}}=F_{E_{2},E_{1}}$. We fix once and for all an orientation on
$F=F_{E_{1},E_{2}}$ by setting its outer normal vector $n$ to be
$n=n_{F}=n_{F}(x)\coloneqq n_{E_{1}}(x)$ for $x\in F$ where $n_{E_{1}}$ is the
outer unit normal field on $\partial E_{1}$. The flux matrix in normal
direction on the face $F$ will be denoted by

$A_{F}\coloneqq(n_{F})_{1}A_{1}+(n_{F})_{2}A_{2}=O_{F}\Lambda_{F}O_{F}^{t}$

where $O_{F}\Lambda_{F}O_{F}^{t}$ is an eigenvalue decomposition of the matrix
$A_{F}$ with $\Lambda_{F}$ being a diagonal matrix and $O_{F}$ being an
orthonormal matrix. Based on this, we define matrices which encode the flux
directions as

$\displaystyle A_{F}^{+}=O_{F}\Lambda_{A}^{+}O^{t}_{F},\quad
A_{F}^{-}=O_{F}\Lambda_{F}^{-}O^{t}_{F},\quad\text{with
}(\Lambda_{F}^{\pm})_{i,i}=\frac{1}{2}\big{(}|(\Lambda_{F})_{i,i}|\pm(\Lambda_{F})_{i,i}\big{)}.$

Note that $A_{F}=A_{F}^{+}+A_{F}^{-}.$ We also introduce a generalization of
the absolute value for such flux matrices by $|A_{F}|=A_{F}^{+}-A_{F}^{-}.$

An element $v_{h}\in\mathcal{V}_{h}$ is multi-valued on any internal face
$F\in\mathcal{F}^{\text{int}}_{h}$. We define its average and jump by

$\left\\{\mskip-5.0mu\\{\mskip-2.0muv_{h}\right\\}\mskip-5.0mu\\}\mskip-2.0mu:=\frac{1}{2}(v_{h}\big{|}_{E_{1}}+v_{h}\big{|}_{E_{2}}),\quad\left\llbracket
v_{h}\right\rrbracket:=v_{h}\big{|}_{E_{1}}-v_{h}\big{|}_{E_{2}}.$

For exterior faces $F\in\mathcal{F}^{\text{ext}}_{h}$ we simply choose the
unit outer normal and extend the definition of jump and average appropriately.

The (unstabilized) upwind semi-discretization in space is then given as: Find
$u_{h}(t)\in\mathcal{V}_{h}$ such that

$(\partial_{t}u_{h}(t),v_{h})_{L^{2}(\Omega)}+a^{\text{upw}}_{h}(u_{h}(t),v_{h})+l_{h}(v_{h})=0\quad\forall\>v_{h}\in\mathcal{V}_{h}$
(1)

with

$\displaystyle a^{\text{upw}}_{h}(u_{h},v_{h})$
$\displaystyle=\sum_{F\in\mathcal{F}_{h}^{\text{int}}}\int_{F}\langle
A_{F}\left\\{\mskip-5.0mu\\{{u_{h}}\right\\}\mskip-5.0mu\\},\left\llbracket{v_{h}}\right\rrbracket\rangle+\frac{1}{2}\langle|A_{F}|\left\llbracket{u_{h}}\right\rrbracket,\left\llbracket{v_{h}}\right\rrbracket\rangle
ds$ $\displaystyle+\sum_{F\in\mathcal{F}_{h}^{\text{ext}}}\int_{F}\langle
A_{F}^{+}u_{h},v_{h}\rangle ds,$ $\displaystyle l_{h}(v_{h})$
$\displaystyle=-\sum_{F\in\mathcal{F}_{h}^{\text{ext}}}\int_{F}\langle
A_{F}^{-}g,v_{h}\rangle ds.$

Here, $\langle\cdot,\cdot\rangle$ denotes the standard scalar product in
$l^{2}$. While we use an upwind flux here, we want to note that our following
construction for the stabilization terms also holds for the Lax-Friedrichs
flux, where some terms cancel. We skipped this for brevity. Discretization in
time is then accomplished by the explicit Euler scheme. If the time step
choice does not reflect the size of smaller cut-cells, this causes stability
issues, which is why we need stabilization terms.

##  3 Stabilization

The main idea of the DoD stabilization is to extend the numerical domain of
dependence of the neighbor cells of a small cut-cell by introducing additional
numerical fluxes on the cut-cell boundary depending on the extrapolated
solutions of the neighbor elements.

Let $\mathcal{I}\subset\mathcal{M}_{h}$ be the set of cut-cells which are
supposed to be stabilized (in practise this set will be chosen based on a
volume fraction with respect to the background cells). For simplicity we
assume that cells in $\mathcal{I}$ are not neighbors of each other. The
$P^{0}$ stabilization fluxes are collected in a term

$J^{0}_{h}(u_{h},v_{h})=\sum_{E\in\mathcal{I}}J^{0,E}_{h}(u_{h},v_{h})$

and the stabilized scheme reads: Find $u_{h}(t)\in\mathcal{V}_{h}$ such that
for any $v_{h}\in\mathcal{V}_{h}$

$(\partial_{t}u_{h}(t),v_{h})_{L^{2}(\Omega)}+a_{h}(u_{h}(t),v_{h})+J^{0}_{h}(u_{h}(t),v_{h})+l_{h}(v_{h})=0.$

$F_{1}$$E_{1}$$F_{2}$$E_{2}$$F_{3}$$E_{3}$$E$
$\int_{F_{3}}\langle({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(\omega_{F_{1}}A_{F_{3}}^{+})_{\text{sym}}(\mathcal{L}^{\text{ext}}_{E_{1}}(u_{h})-u_{h}^{E_{\text{cut}}})}\\\
\phantom{+}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(\omega_{F_{2}}A_{F_{3}}^{+})_{\text{sym}}(\mathcal{L}^{\text{ext}}_{E_{2}}(u_{h})-u_{h}^{E_{\text{cut}}})}\\\
\phantom{+}+{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}(\omega_{F_{3}}A_{F_{3}}^{+})_{\text{sym}}(\mathcal{L}^{\text{ext}}_{E_{3}}(u_{h})-u_{h}^{E_{\text{cut}}})}\rangle\llbracket
v\rrbracket ds$ Figure 2: DoD stabilization for the acoustic equation on a
triangular cut-cell. On the left are the stabilization fluxes on the face
$F_{3}$ (without the parameter $\eta_{E}$), colored in correspondence with the
arrows on the right indicating the different couplings. Note that there will
be additional stabilization fluxes on $F_{1}$ and $F_{2}$. The green flux is
somewhat of a curiosity and might just be a product of the way we construct
our weighting matrices for the specific case of the acoustic equation. In a
sense it acts similar to a reflecting wall.

For any $E\in\mathcal{M}_{h}$ we introduce a map
$\mathcal{L}^{\text{ext}}_{E}:\mathcal{V}_{h}(E)\to\mathcal{P}^{0}(\Omega)^{3}$,
called an extension operator, such that
$\mathcal{L}^{\text{ext}}_{E}(u_{h})\big{|}_{E}=u_{h}\big{|}_{E}$. The cell
stabilization terms $J^{0,E}_{h}(u_{h},v_{h})$ then contain extended fluxes

$\eta_{E}\\!\\!\sum_{(E_{1},E_{2})\in\mathcal{N}(E)}\int_{F_{E,E_{2}}}\\!\\!\Big{\langle}(\omega_{F_{E,E_{1}}}A_{F_{E,E_{2}}}^{+})_{\text{sym}}(\mathcal{L}^{\text{ext}}_{E_{1}}(u_{h})-u_{h}\big{|}_{E}),\llbracket
v_{h}\rrbracket\Big{\rangle}ds,$

with appropriate weighting matrices
$\omega_{F_{E,E_{1}}}\in\mathbb{R}^{3\times 3}$ and a penalty parameter
$\eta_{E}\in(0,1]$. Here, we have denoted the symmetrization
$\frac{1}{2}(A+A^{t})$ for a matrix $A\in\mathbb{R}^{3\times 3}$ by
$A_{\text{sym}}$. This introduces a direct mass transport from $E_{1}$ to
$E_{2}$ for any pair $(E_{1},E_{2})$ of neighbors of a small cut-cell $E$.
(For a triangular cut-cell, the sum contains 9 terms.) This is a result of the
inherent nature of the wave equation, creating waves in all possible
directions. The weighting matrices describe how much of an inflow coming from
$E_{1}$ is transported to $E_{2}$. Figure 2 shows an illustration. Similarly
to what was proposed in [4], we require for the weighting matrices that

$\displaystyle\sum_{E_{2}\in\mathcal{N}(E)}\int_{F_{E,E_{2}}}\omega_{E,E_{1}}A^{+}_{F_{E,E_{2}}}ds$
$\displaystyle=-\int_{F_{E,E_{1}}}A^{-}_{F_{E,E_{1}}}ds\qquad\forall\>E_{1}\in\mathcal{N}(E),$
(2a) $\displaystyle\sum_{E_{1}\in\mathcal{N}(E)}\omega_{F_{E,E_{1}}}$
$\displaystyle=\,\text{Id}_{3\times 3}.$ (2b)

The first equation describes how incoming flow is distributed among the cut-
cell’s neighbors while the second equation ensures that the overall amount of
flow over a single face is preserved.

In [4] we were able to prove a discrete dissipation and $L^{2}$-stability
result. One key step of the proof is an application of the binomial formula
relying on the symmetry of the matrices
$\omega_{F_{E,E_{1}}}A_{F_{E,E_{2}}}^{+}$, which holds given simultaneously
diagonalizable (equivalently, commuting) system matrices. For the acoustic
equation this matrix product will in general not be symmetric and the proof
breaks down. A working fix is to instead take the symmetrization of the matrix
$\omega_{F_{E,E_{1}}}A_{F_{E,E_{2}}}^{+}$.

Taking the symmetrization is not enough by itself though since the resulting
flux matrices potentially possess negative eigenvalues leading to incorrect
fluxes. This can be corrected by introducing terms

$-\kappa\eta_{E}\\!\\!\sum_{(E_{1},E_{2})\in\mathcal{N}(E)}\int_{F_{E,E_{2}}}\\!\\!\Big{\langle}(\omega_{F_{E,E_{1}}}A_{F_{E,E_{2}}}^{+})_{\text{sym}}^{-}(\mathcal{L}^{\text{ext}}_{E_{1}}(u_{h})-u_{h}\big{|}_{E_{2}}),\mathcal{L}^{\text{ext}}_{E_{1}}(v_{h})-v_{h}\big{|}_{E_{2}}\Big{\rangle}ds.$

For any $\kappa\geq 1$ this term ensures $L^{2}$-stability of the semi-
discrete form. We observe that certain choices of $\kappa$ significantly
improve the numerical approximation; the optimal choice still needs to be
investigated. Note that this new term has an opposite sign, uses the negative
part of the symmetrized form, and subtracts the solution from cell neighbor
$E_{2}$ (instead of the cut-cell $E$).

Putting those components together leads to a cell stabilization term

$\displaystyle J^{0,E}_{h}$ $\displaystyle(u_{h},v_{h})$
$\displaystyle=\eta_{E}\\!\\!\sum_{(E_{1},E_{2})\in\mathcal{N}(E)}\int_{F_{E,E_{2}}}\\!\\!\Big{\langle}(\omega_{F_{E,E_{1}}}A_{F_{E,E_{2}}}^{+})_{\text{sym}}(\mathcal{L}^{\text{ext}}_{E_{1}}(u_{h})-u_{h}\big{|}_{E}),\llbracket
v_{h}\rrbracket\Big{\rangle}ds$
$\displaystyle-\kappa\eta_{E}\\!\\!\sum_{(E_{1},E_{2})\in\mathcal{N}(E)}\int_{F_{E,E_{2}}}\\!\\!\Big{\langle}(\omega_{F_{E,E_{1}}}A_{F_{E,E_{2}}}^{+})_{\text{sym}}^{-}(\mathcal{L}^{\text{ext}}_{E_{1}}(u_{h})-u_{h}\big{|}_{E_{2}}),\mathcal{L}^{\text{ext}}_{E_{1}}(v_{h})-v_{h}\big{|}_{E_{2}}\Big{\rangle}ds.$

For this formulation together with the properties (2a) and (2b) one can prove
$L^{2}$-stability for the semi-discretization in space. Unfortunately the
proof is too long for this contribution to be included.

The weighting matrices can be directly computed from (2a). Due to the system
matrices being divergence-free, this equation is equivalent to

$\omega_{F_{E,E_{1}}}\Big{(}\sum_{E_{2}\in\mathcal{N}(E)}|F_{E,E_{2}}|A_{F_{E,E_{2}}}^{-}\Big{)}=|F_{E,E_{1}}|A_{F_{E,E_{1}}}^{-}.$

The matrix $(\sum_{E_{2}\in\mathcal{N}(E)}|F_{E,E_{2}}|A_{F_{E,E_{2}}}^{-})$
is invertible since on a triangular cut-cell we got three summands, each of
rank one, and the three column spaces are linearly independent. Thus we have

$\omega_{F_{E,E_{1}}}=|F_{E,E_{1}}|A_{F_{E,E_{1}}}^{-}\Big{(}\sum_{E_{2}\in\mathcal{N}(E)}|F_{E,E_{2}}|A_{F_{E,E_{2}}}^{-}\Big{)}^{-1}.$

##  4 Numerical results

We present numerical results to support our findings. Our implementation uses
the DUNE framework [1, 2], in particular the dune-udg module [3, 5] and the
TPMC library [7]. We consider an analytic test case [9] with

$u(x,y,t)=\frac{1}{c}\begin{pmatrix}-\cos(2\pi ct)(\sin(2\pi x)+\sin(2\pi
y))\\\ \sin(2\pi ct)\cos(2\pi x)\\\ \sin(2\pi ct)\cos(2\pi y)\end{pmatrix}$

Initial condition $u_{0}$ and inflow boundary conditions $g$ are given by the
exact solution. The speed of sound is chosen as $c=\frac{1}{2}$, the final
time is $T=0.3$. For the CFL-condition we choose $\Delta t=0.3\frac{\Delta
x}{c}$ where $\Delta x=\frac{1}{N}$ with $N$ being the number of fundamental
cells in one direction of the grid. We choose values for $N$ from the set
$\\{400,..,1200\\}$, starting at $400$ and with a stepsize of $23$. In our
experiments the volume fractions of cut-cells are in the interval $[7.24\cdot
10^{-10},5.37\cdot 10^{-5}]$. For the cut angle we set $\gamma=35^{\circ}$ and
choose $x_{0}=0.2001$ for the start of the cut. We classify a cell
$E\in\mathcal{M}_{h}$ as a small cut-cell if $\frac{|E|}{(1/N)^{2}}\leq 0.4$.
The penalty parameter $\eta_{E}$ is chosen as $(1-\eta_{E})=\frac{|E|}{\Delta
tc\max_{F\subset\partial E}|F|}$.

We compute the $L^{2}$-error of the discrete solution at the final time $T$
with respect to the exact solution. We also compute the pointwise error at
certain quadrature points, which gives an approximation to the error in the
$L^{\infty}$-norm. We will display results for $\kappa\in\\{1.0,7.5\\}$ to
display the effect of this additional parameter.

The computed errors are plotted in figure 3. We observe the expected
convergence order in the $L^{2}$-norm in all components, regardless of the
choice of $\kappa$. For the $L^{\infty}$-norm the situation differs. The
pressure component shows again optimal convergence behavior, independent of
$\kappa$. However for the velocity components the convergence order is
influenced by the choice of $\kappa$ and we obtain better absolute errors and
a better convergence order (at the cost of some mild wiggles) for a larger
$\kappa=7.5$. The wiggles suggest that the parameter $\kappa$ should be chosen
cell dependent, but how to choose it exactly requires further investigation.

Figure 3: Error at $T=0.3$ in the $L^{2}$ norm (left) and in the $L^{\infty}$
norm (right). The straight purple line is for reference and denotes
convergence of order 1.

##  5 Discussion and Outlook

We presented an extension of the DoD stabilization to the acoustic wave
equation for $P^{0}$ trial and test functions on a structured grid with
triangular cut-cells. We observed a decent convergence behavior in our
numerical test setup. We note that the new parameter $\kappa$ included in the
extension has a considerable influence on the measured error. Its precise
effect and optimal choice will be investigated more in the future. An
extension to higher-order approximations, as well as to non-linear systems, e.
g. the Euler equations, is ongoing research.

###  5.0.1 Acknowledgements

The authors acknowledge support by the Deutsche Forschungsgemeinschaft as
project 439956613 under contract numbers EN 1042/5-1 and MA 7773/4-1/2, as
well as under Germany’s Excellence Strategy EXC 2044
390685587, Mathematics Münster: Dynamics – Geometry – Structure.

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