A nite dierence - discontinuous Galerkin method for the wave equation in second order form Siyang WangGunilla Kreiss

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A ﬁnite diﬀerence - discontinuous Galerkin method for the wave

equation in second order form

Siyang Wang∗Gunilla Kreiss†

Abstract

We develop a hybrid spatial discretization for the wave equation in second order form,

based on high-order accurate ﬁnite diﬀerence methods and discontinuous Galerkin methods.

The hybridization combines computational eﬃciency of ﬁnite diﬀerence methods on Cartesian

grids and geometrical ﬂexibility of discontinuous Galerkin methods on unstructured meshes.

The two spatial discretizations are coupled by a penalty technique at the interface such that

the overall semidiscretization satisﬁes a discrete energy estimate to ensure stability. In addition,

optimal convergence is obtained in the sense that when combining a fourth order ﬁnite diﬀerence

method with a discontinuous Galerkin method using third order local polynomials, the overall

convergence rate is fourth order. Furthermore, we use a novel approach to derive an error

estimate for the semidiscretization by combining the energy method and the normal mode

analysis for a corresponding one dimensional model problem. The stability and accuracy analysis

are veriﬁed in numerical experiments.

Keywords: ﬁnite diﬀerence methods, discontinuous Galerkin methods, hybrid methods, wave

equations, normal mode analysis

AMS: 65M06, 65M12

1 Introduction

Second order hyperbolic partial diﬀerential equations describe wave-dominated problems, for ex-

ample the acoustic wave equation, the elastic wave equation and Einstein’s equations of general

relativity. In realistic models, waves propagate over long time in large domains with heterogeneous

material properties and complex geometries. As a result, analytical solutions can generally not be

derived. Numerical simulation is a powerful alternative to seek an approximated solution to the

governing equations. For time-dependent problems, it is important to use stable numerical methods

that do not allow unphysical growth in the numerical solution. In addition, by the classical disper-

sion analysis [16, 20], high-order accurate numerical methods are more computationally eﬃcient

than low-order methods when the solution is suﬃciently smooth. Over the years, there has been

extensive work on stable and high-order numerical methods for wave propagation problems.

The ﬁnite diﬀerence (FD) method is conceptually simple, computationally eﬃcient and easy

to implement. Traditionally, it was challenging to derive stable and high-order FD discretizations

for hyperbolic problems. This challenge has partly been overcome by using FD stencils with a

summation-by-parts (SBP) property [21], in combination with the simultaneous-approximation-

term (SAT) technique [4] to impose boundary conditions. The integration-by-parts principle is

∗Department of Mathematics and Mathematical Statistics, Ume˚a University, Ume˚a, Sweden. Email:

siyang.wang@umu.se

†Division of Scientiﬁc Computing, Department of Information Technology, Uppsala University, Uppsala, Sweden.

arXiv:2210.13577v1 [math.NA] 24 Oct 2022

the key ingredient to derive continuous energy estimates for the PDEs. The SBP-SAT method-

ology mimics the integration-by-parts principle for a discrete energy estimate to ensure that the

semidiscretization is stable. The relation between the SBP-SAT FD method and the discontinuous

Galerkin spectral element method is investigated in [10].

The FD method in its basic form is only applicable to problems on rectangular-shaped domains.

For other shapes, a curvilinear grid based on coordinate transformation is used to resolve geomet-

rical features [29]. In general, the computational domain cannot be easily mapped to a reference

domain. In this case, we decompose the computational domain into subdomains and use a multi-

block FD approach. The multiblock SBP-SAT methods on curvilinear grid have been derived for

the wave equation [31] and the elastic wave equation [6] in second order form. This approach works

well on nearly Cartesian grids but is not suitable in many realistic models with complex geometry,

because it is diﬃcult to ﬁnd a smooth coordinate transformation.

Recently, there have been eﬀorts in hybridizing the FD discretization with a Galerkin method

on unstructured meshes so that the overall discretization is both computationally eﬃcient and

geometrically ﬂexible. The main diﬃculty originates from the fact that the two discretizations

have diﬀerent discrete l2inner product. This scenario also occurs at an FD-FD discretization with

diﬀerent grid sizes, i.e. nonconforming grid interfaces. For the wave equation in ﬁrst order form,

SBP-preserving interpolation operators are constructed in [23] for an FD-FD nonconforming inter-

face with grid size ratio 1:2. With an SBP operator of interior order 2p, the observed convergence

rate in numerical experiments is p+ 1, which is the same as a multiblock FD with only conforming

grid interfaces. In [19], the SBP FD is coupled with the discontinuous Galerkin (DG) method by

using a projection technique that preserves the SBP property and the semidiscretization satisﬁes

an energy estimate. With an SBP operator of interior order 2pand the DG method based on local

polynomials of degree p, the observed convergence rate in numerical experiments is p+ 1. There

has also been important work on the hybridization of the SBP FD discretization with the ﬁnite

element method for the isotropic elastic wave equation [9] and the conservation law [5], with a focus

on stability rather than accuracy.

In this paper, we consider the wave equation in second order form. Comparing to ﬁrst order

form, solving the wave equations in second order form has advantages. There are fewer unknown

variables, thus requiring less computation and memory storage. In addition, when imposing the

boundary and interface conditions properly, the SBP FD discretization based on operators of inte-

rior order 2pcan converge to order p+ 2 , i.e. one order higher than solving the same equation in

ﬁrst order form. However, it is challenging to solve the wave equations in second order form from

both stability and accuracy aspects. A generalization of the interpolation technique from [23] to

the wave equation in second order form converges only to suboptimal order p+ 1. For stability, an

additional norm-contraction constraint on the interpolation operators is required. This additional

constraint is removed by using a new SAT technique [32], which does not simultaneously improve

the accuracy property. In [1], the optimal convergence rate p+ 2 is recovered by using two pairs of

order-preserving interpolation operators.

The ﬁrst contribution of this paper is an FD-DG spatial discretization for the wave equation

in two space dimension in second order form. We construct novel projection operators to combine

the SBP FD discretization with the symmetric interior penalty discontinuous Galerkin (IPDG)

method [13]. The overall discretization satisﬁes a discrete energy estimate to guarantee stability.

In addition, the FD-DG discretization converges to the optimal order in the sense that with SBP

operators of interior order four and the IPDG based on local polynomials of degree three, the

observed convergence rate is four.

Our second contribution is a new framework for the accuracy analysis of the FD-DG discretiza-

tion. A priori error estimates for the DG discretization are often derived by the energy method

using special projection operators and approximation theory [17], whereas sharp error estimates

for the FD discretization is derived by the normal mode analysis in Laplace space [14, 15]. Though

both are well-established, they are two distinct approaches. To analyze the accuracy of the FD-DG

discretization, we consider the wave equation in one space dimension and cast the DG scheme

into matrix form, and realize its components as diﬀerence stencils. It is well-known that the re-

sulting DG truncation error indicates a suboptimal convergence rate. By a careful analysis of the

truncation error in the discrete norm associated with the DG discretization, we obtain sharp error

estimates by the energy method for the DG discretization. After that, we combine it with the

normal mode analysis for the FD-DG interface treatment and obtain an optimal convergence rate

for the overall discretization.

The rest of the paper is organized as follows. In Sec. 2, we introduce an FD-DG spatial

discretization for the wave equation in one space dimension. After that, we present our novel

approach for deriving an apriori error estimate for the hybridization. In Sec. 3, we start with

projection operators that are used in the numerical scheme for the wave equation in two space

dimension. We then analyze the stability property of the overall discretization by deriving a discrete

energy estimate. Numerical examples are presented in Sec. 4 to verify the theoretical results. In

the end, we draw conclusion in Sec. 5.

2 Spatial discretization in 1D and error analysis

In this section, we start by introducing the concept of SBP and its important properties, and

deriving an FD-DG spatial discretization of the wave equation in one space dimension. After that,

we present a novel approach for accuracy analysis and derive an a priori error estimate for the

FD-DG semidiscretization.

2.1 Summation-by-parts ﬁnite diﬀerence operators

Consider a bounded interval Ithat is discretized by a uniform grid xi, i = 1,2,··· , n with grid

spacing h. Let f, g ∈C∞(I) and deﬁne the grid functions fi=f(xi), gi=g(xi), and vectors

f= [f1, f2,··· , fn]T,g= [g1, g2,··· , gn]T.

We also deﬁne the standard L2inner product (f, g)I=RIfgdx, and a discrete l2norm kfk=

phPn

i=1 |fi|2.

Next, we consider the ﬁnite diﬀerence approximation of the second derivative, D≈d2

dx2. The

SBP property of Dis deﬁned as follows [26].

Deﬁnition 1 (second derivative SBP property) The ﬁnite diﬀerence operator D≈d2

dx2is a

second derivative SBP operator if it can be written as

D=H−1(−A+endT

n−e1dT

1),(1)

where en= [0,0,··· ,0,1]Tand e1= [1,0,··· ,0]T. The ﬁrst derivative approximations are dT

1f≈

dx (x1)and dT

nf≈df

dx (xn). The operator His symmetric positive deﬁnite, and Ais symmetric

positive semideﬁnite.

The operator Hdeﬁnes a discrete inner product and norm, and is also a quadrature [18].

Similarly, the operator Adeﬁnes a discrete semi-norm. They satisfy the relations,

fTHg≈Zxn

fgdx, fTAg≈Zxn

dxdx.

We recognize Hand Aas the mass and stiﬀness matrix for a Galerkin method.

In the interior, the SBP operators Dare based on standard central ﬁnite diﬀerence stencils with

truncation error O(h2p). On a few grid points near boundaries, one-sided stencils are used to satisfy

the SBP property. When His diagonal, the truncation error of the one-sided boundary stencil can

at best be O(hp). The truncation error of the ﬁrst derivative approximation at the boundaries is

O(hp+1). We denote the order of accuracy of Das (2p, p). The SBP property of (1) can also be

written as

gTHDf=−gTAf+gTendT

nf−gTe1dT

1f,

which is a discrete analogue of the integration-by-parts formula,

Zxn

gfxxdx =−Zxn

gxfxdx +g(xn)fx(xn)−g(x1)fx(x1).

A so-called borrowing technique of the SBP operator Dis important for proving stability for

certain problems, such as the wave equation with Dirichlet boundary conditions [25] and material

interface conditions [24]. It is also used to derive an energy estimate for a dual-consistent dis-

cretization of the heat equation [7]. The borrowing capacity for the borrowing technique is deﬁned

as follows.

Deﬁnition 2 (borrowing capacity) The borrowing capacity is the maximum value of β > 0such

that ˜

A=A−βh(d1dT

1+dndT

is symmetric positive semideﬁnite. Here, his the grid spacing, d1and dnare the same ﬁrst

derivative operators as in (1).

Remark 1 The borrowing capacity depends on the order of accuracy of the SBP operator but does

not depend on h. For the precise values of the borrowing capacity, see [8, 24, 25]. The borrowing

technique is a ﬁnite diﬀerence analogue to using the inverse inequality to derive estimates for ﬁnite

element methods. To see this relation, we write

fTAf−βhfT(d1dT

1+dndT

n)f=fT˜

Af≥0,

which leads to

fTAf≥βh((dT

1f)2+ (dT

nf)2).

Recalling fTAf≈Rxn

x1(df

dx )2dx,dT

1f≈df

dx (x1), and dT

nf≈df

dx (xn)the above relation is a discrete

analogue of the inverse inequality [3].

2.2 An FD-DG discretization in 1D

An SBP operator only approximates a derivative but does not impose any boundary condition.

When solving an initial-boundary-value problem, the SAT technique is often used to impose bound-

ary and interface conditions weakly. The main idea of SAT is to add penalty terms in the semidis-

cretization such that a discrete energy estimate can be obtained. For accuracy, it is important that

the penalty terms converge to zero as the mesh size goes to zero. The SBP-SAT discretization for

the wave equation in second order form was derived for various boundary conditions [2, 25, 26] and

material interface conditions [24].

In the IPDG method [13], boundary and material interface conditions are naturally imposed by

using numerical ﬂuxes. In the following, we use the wave equation in one space dimension as the

Figure 1: An FD grid and DG elements in one space dimension.

model problem, and derive a stable FD-DG semidiscretization. In this case, the interface between

the two semidiscretizations is only a point in space and the numerical treatment does not involve

the diﬃculties for higher dimensional problems. Nonetheless, the scheme and stability analysis for

the one dimensional model problem demonstrate the penalty technique to combine the FD and DG

semidiscretizations and prepare for the accuracy analysis afterwards.

For the analysis, we consider

Utt =Uxx, x ∈(−∞,∞), t ∈(0, T ],

with smooth initial conditions with bounded support. We discretize the equation in space by the

SBP FD method in x∈(−∞,0), and the IPDG in x∈(0,∞). At the FD-DG interface at x= 0,

we impose the interface conditions U(0−, t) = U(0+, t) and Ux(0−, t) = Ux(0+, t) weakly.

We discretize the FD domain (−∞,0) by a uniform grid xj=−(j−1)h, where j= 1,2,3,··· and

his the grid spacing. In the DG domain, we partition (0,∞) into disjoint elements Ij= (Xj, Xj+1)

with j= 1,2,3,···. For simplicity, we assume that the elements have equal length such that

Xj= (j−1)h, j = 1,2,3,···. We note that the points x1and X1coincide at the FD-DG interface,

see Figure 1. We also note that the degrees of freedom (DOFs) are duplicated on the inter-element

interfaces Xj, j = 2,3,···, on the DG side.

2.3 Stability of the FD-DG discretization in 1D

The FD discretization can be written as

wtt =H−1(−A+endT

n)w

−1

2H−1en(dT

nw−u(1)

xΓ) + 1

2H−1dn(eT

nw−u(1)

Γ)−τ

hH−1en(eT

nw−u(1)

Γ),(2)

where w= [w1, w2,···]Tis the ﬁnite diﬀerence solution, wj≈U(xj, t), j = 1,2,···. On the right-

hand side, the ﬁrst term is the approximation of Uxx, and the last three terms impose weakly the

interface conditions. More precisely, the second term imposes continuity of Ux, and the third and

fourth terms impose weakly continuity of U. The terms u(1)

Γand u(1)

xΓare the DG solution and

its derivative at the interface, i.e. u(1)

Γ=u(1)(X1, t) and u(1)

xΓ=u(1)

x(X1, t). We note that (2) is a

generalization of the SBP-SAT scheme for the 1D wave equation with a material interface [24].

For the DG solution, for every ﬁxed time we seek solution in the following space

h={v:v|Ij∈Pk(Ij), j = 1,2,···},(3)

where Pk(Ij) denotes the space of polynomials of degree at most kin Ij. The DG discretization

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Anitedierence-discontinuousGalerkinmethodforthewaveequationinsecondorderformSiyangWang*GunillaKreissAbstractWedevelopahybridspatialdiscretizationforthewaveequationinsecondorderform,basedonhigh-orderaccuratenitedierencemethodsanddiscontinuousGalerkinmethods.Thehybridizationcombinescomputationale...

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A nite dierence - discontinuous Galerkin method for the wave equation in second order form Siyang WangGunilla Kreiss

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