Non-conforming interface conditions for the second-order wave equation Gustav Eriksson

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Non-conforming interface conditions for the

second-order wave equation

Gustav Eriksson

Department of Information Technology, Uppsala University, PO

Box 337, Uppsala, S-751 05, Sweden.

Contributing authors: gustav.eriksson@it.uu.se;

Abstract

Imposition methods of interface conditions for the second-order wave

equation with non-conforming grids is considered. The spatial discretiza-

tion is based on high order ﬁnite diﬀerences with summation-by-parts

properties. Previously presented solution methods for this problem,

based on the simultaneous approximation term (SAT) method, have

shown to introduce signiﬁcant stiﬀness. This can lead to highly inef-

ﬁcient schemes. Here, two new methods of imposing the interface

conditions to avoid the stiﬀness problems are presented: 1) a pro-

jection method and 2) a hybrid between the projection method and

the SAT method. Numerical experiments are performed using tra-

ditional and order-preserving interpolation operators. Both of the

novel methods retain the accuracy and convergence behavior of the

previously developed SAT method but are signiﬁcantly less stiﬀ.

Keywords: Summation-by-parts, High order, Non-conforming interface,

Projection

1 Introduction

It is well known that high order ﬁnite diﬀerences are highly eﬃcient for large-

scale wave propagation problems [1]. However, the design of such schemes

requires particular care at the boundaries to obtain stability. One way to obtain

stable high order ﬁnite diﬀerence schemes is to use ﬁnite diﬀerence operators

with a summation-by-parts (SBP) property together with simultaneous-

approximation-terms (SBP-SAT) [2], the projection method (SBP-P) [3,4] or

arXiv:2210.13115v1 [math.NA] 24 Oct 2022

2Non-conforming interface conditions for the second-order wave equation

ghost points (SBP-GP) [5]. SBP ﬁnite diﬀerence operators are essentially stan-

dard ﬁnite diﬀerence stencils in the interior with boundary closures carefully

designed to mimic integration by parts in the discrete setting. The SBP diﬀer-

ence operators have an associated discrete inner product such that a discrete

energy equation that is analogous to the continuous equation can be derived.

The boundary conditions should be imposed such that the scheme exhibits no

non-physical energy growth, sometimes referred to as strict stability [6]. The

SAT method achieves this by adding penalty terms that weakly impose the

boundary conditions such that the resulting scheme is stable, see for example

[7]. The SBP-GP method adds ghost points at the boundaries and computes

their values such that the boundary conditions are imposed and the scheme

is stable [8,9]. The projection method derives an orthogonal projection and

rewrites the problem such that it is solved in the subspace of solutions where

the boundary conditions are exactly fulﬁlled, see [10].

An important aspect of ﬁnite diﬀerence methods is the ability to split the

computational domain into blocks and couple them across the interfaces. This

is necessary to handle complex geometries, but also to increase the eﬃciency

of the schemes. For example, in the case of the wave equation, a ﬁner grid

spacing is only needed in regions of the domain where the wave speed is high.

In other regions, a coarser grid may be used. In general, the grid points at each

side of an interface are non-conforming, in which case interpolations are used

to couple the solutions. In the framework of SBP ﬁnite diﬀerences, it is crucial

that the method of imposing the interpolated interface conditions preserves

the SBP properties of the diﬀerence operators.

The construction of interpolation operators along with SATs to obtain sta-

ble schemes with non-conforming interfaces has received signiﬁcant attention in

the past [11–13]. In [11] so-called SBP-preserving interpolation operators (here

referred to as norm-compatible) were ﬁrst constructed and used to derive stable

schemes for general hyperbolic and parabolic problems. However, it was noted

in [13,14] that the global convergence rate was decreased by one (compared

to the convergence rate with conforming grids) for problems involving sec-

ond derivatives in space. In [15] this is solved by constructing order-preserving

(OP) interpolation operators along with SATs such that the global convergence

rate is preserved. The new operators come in two norm-compatible pairs (a

pair consists of one restriction operator and one prolongation operator), where

one of the operators in each pair is of one order higher accuracy. Using both

pairs, an SAT is presented in [15] where the ﬁrst interface condition (continu-

ity of the solution) is imposed using the accurate interpolation and the second

interface condition (continuity of the ﬁrst derivative) using the less accurate

interpolation.

A major downside of the SBP-SAT discretizations is the necessary decom-

position of the second derivative SBP operator to obtain an energy estimate.

Often referred to as the ”borrowing trick” [16]. This procedure is known to

introduce additional stiﬀness to the problem, especially for large wave speed

discontinuities. The main contribution of the current work is two new methods

Non-conforming interface conditions for the second-order wave equation 3

avoiding this problem, one using SBP-P and the other a hybrid SBP-P-SAT.

The analysis and numerical experiments are done on the second-order wave

equation. However, the discrete Laplace operators presented are equally appli-

cable to the heat equation and the Schr¨odinger equation. There are indications

that the new methods can be applied to other problems, such as ﬁrst-order

hyperbolic systems, but this is out of the scope of the current work.

The paper is structured as follows: In Section 2some necessary deﬁni-

tions and the discrete operators are introduced. In Section 3the continuous

problem is presented. The new semi-discrete schemes are presented in Section

4. The time discretization is presented in Section 5. In Section 6numerical

experiments validating the new methods and comparing them to the SBP-SAT

schemes are presented. Conclusions are drawn in Section 7.

2 Deﬁnitions

Let

(u, v) = ZΩ

uv dx and kuk2= (u, u),(1)

deﬁne an inner product and the corresponding norm for functions u, v on a

rectangular domain Ω. The domain is split across the x-axis into a left and a

right block, denoted ΩLand ΩR. The two blocks are discretized using m(u,v)

and m(u,v)

yequidistant grid points in the x- and y-directions respectively.

The second-derivatives in each block and direction are approximated using

one-dimensional SBP ﬁnite diﬀerence operators [17] satisfying

D2=H−1(−M+erd>

r−eld>

l),(2)

where His diagonal and positive deﬁnite, Mis symmetric and positive semi-

deﬁnite, e>

l,r are row-vectors extracting the solution at the ﬁrst and last grid

points and d>

l,r are row-vectors approximating the ﬁrst derivative of the solu-

tion at the ﬁrst and last grid points. The matrix D2is referred to as a

2pth-order accurate second derivative SBP operator. In the interior D2con-

sists of a 2porder accurate central ﬁnite diﬀerence stencil. On the boundaries,

for the SBP properties to hold with a diagonal H, the order of accuracy is lim-

ited to p. Thus, the theoretical global order of accuracy with these operators

is min(2p, p+2) [18]. In this paper, numerical results are presented for 4th and

6th order SBP operators. Hence, the expected convergence rates are 4 and 5.

The matrix Hdeﬁnes a one-dimensional discrete inner product and norm

(u, v)H=u>Hv and kuk2

H= (u, u)H.(3)

4Non-conforming interface conditions for the second-order wave equation

The one-dimensional operators are extended to two dimensions using Kro-

necker products as follows:

D2x= (D2⊗Imy), D2y= (Imx⊗D2),

Hx= (H⊗Imy), Hy= (Imx⊗H),

Mx= (M⊗Imy), My= (Imx⊗M),

eW= (e>

l⊗Imy), eE= (e>

r⊗Imy),

eS= (Imx⊗e>

l), eN= (Imx⊗e>

r),

dW= (d>

l⊗Imy), dE= (d>

r⊗Imy),

dS= (Imx⊗d>

l), dN= (Imx⊗d>

r),

(4)

where Imdenotes the m×midentity matrix. The discrete inner product and

norm over the 2D domain is given by

(u, v)¯

H=u>¯

Hv and kuk2

H= (u, u)¯

H,(5)

where ¯

H=HxHy. The discrete Laplace operator is given by

DL=D2x+D2y.(6)

Using the SBP properties (2), the discrete two-dimensional Laplace operator

can be written as

DL=H−1

x(−Mx+e>

EdE−e>

WdW) + H−1

y(−My+e>

NdN−e>

SdS),(7)

or for two vectors u1,2∈Rmxmywe have

(u1, DLu2)¯

H=−u>

1(HyMx+HxMy)u2+ (eEu1, dEu2)H−(eWu1, dWu2)H

+ (eNu1, dNu2)H−(eSu1, dSu2)H.

(8)

In the upcoming analysis, let the solutions in the left block be denoted by u,

and in the right block by v. Superscripts (u) and (v) will be used to denote

which block an operator belongs to. For example, the inner-product matrix

H(u)acts on solution vectors in the left block, with m(u)

xm(u)

yunknowns.

2.1 Interpolation operators

Interpolation operators are used at the interface to couple two blocks with

non-conforming grid points. Let Iu2vdenote the operator interpolating from

left to right, and Iv2uthe operator interpolating from right to left. See Figure

1. For stability, we require that the pair of operators are norm-compatible, i.e.

they must satisfy

(Iv2uv, u)H(u)= (v, Iu2vu)H(v),∀u∈Rm(u)

y, v ∈Rm(v)

y.(9)

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Non-conforminginterfaceconditionsforthesecond-orderwaveequationGustavErikssonDepartmentofInformationTechnology,UppsalaUniversity,POBox337,Uppsala,S-75105,Sweden.Contributingauthors:gustav.eriksson@it.uu.se;AbstractImpositionmethodsofinterfaceconditionsforthesecond-orderwaveequationwithnon-conforming...

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Non-conforming interface conditions for the second-order wave equation Gustav Eriksson

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