The Neumann boundary condition for the two-dimensional Lax-Wendro scheme Antoine Benoit Jean-Fran cois Coulombel

2025-05-06 0 0 560.97KB 33 页 10玖币
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The Neumann boundary condition
for the two-dimensional Lax-Wendroff scheme
Antoine Benoit∗& Jean-Fran¸cois Coulombel†
October 12, 2022
Abstract
We study the stability of the two-dimensional Lax-Wendroff scheme with a stabilizer that approx-
imates solutions to the transport equation. The problem is first analyzed in the whole space in order
to show that the so-called energy method yields an optimal stability criterion for this finite difference
scheme. We then deal with the case of a half-space when the transport operator is outgoing. At the
numerical level, we enforce the Neumann extrapolation boundary condition and show that the cor-
responding scheme is stable. Eventually we analyze the case of a quarter-space when the transport
operator is outgoing with respect to both sides. We then enforce the Neumann extrapolation bound-
ary condition on each side of the boundary and propose an extrapolation boundary condition at the
numerical corner in order to maintain stability for the whole numerical scheme.
AMS classification: 65M12, 65M06, 65M20.
Keywords: transport equations, numerical schemes, domains with corners, boundary conditions, stability.
Notation. For da positive integer and J⊂Zd, we let `2(J;R) denote the Hilbert space of real
valued, square integrable sequences indexed by Jand equipped with the norm:
∀u∈`2(J;R),kuk2
`2(J):= X
j∈J
u2
j.
The corresponding scalar product is denoted h;i`2(J).
1 Introduction
We explore in this article the relevance of extrapolation boundary conditions for outgoing transport
equations in two space dimensions. In one space dimension, a general stability and convergence theory has
been developed in [CL20] following, among others, previous works by Kreiss and Goldberg [Kre66,Gol77].
The main results in [Kre66,Gol77,CL20] assert that, for an explicit,one time step finite difference scheme
that is stable in `2(Z;R) and that is consistent with the transport equation:
∂tv+a ∂xv= 0 , a < 0,(t, x)∈R+×R,
∗Email: antoine.benoit@univ-littoral.fr. Research of the author was supported by ANR project NABUCO, ANR-
17-CE40-0025.
†Institut de Math´ematiques de Toulouse ; UMR5219, Universit´e de Toulouse ; CNRS, F-31062 Toulouse Cedex 9,
France. Email: jean-francois.coulombel@math.univ-toulouse.fr. Research of the author was supported by ANR project
NABUCO, ANR-17-CE40-0025.
1
arXiv:2210.05352v1 [math.NA] 11 Oct 2022
then extrapolation numerical boundary conditions at the origin yield a numerical scheme that is stable in
`2(N;R) (that is, on the half-line). The result is independent of the extrapolation order that is chosen at
the boundary. We refer to [CL20, Theorem 3.1] for a detailed statement. We aim here at understanding
the influence of tangential directions on this stability result in higher space dimension.
Numerical boundary conditions for two-dimensional hyperbolic problems have been investigated, for
instance, in [AG79,Slo83], by the so-called normal mode analysis. This method is, to some extent, optimal
to characterize stability but it usually leads to rather involved algebraic calculations that, sometimes,
cannot be carried out. In this article, we rather wish to developed an energy method in order to deal
with more involved geometries as the quarter-space. We focus on a second-order discretization of the
transport equation that was originally proposed by Lax and Wendroff [LW64]. This approximation does
not rely on any dimensional splitting, which prevents us from using one-dimensional arguments, and it is
second order accurate with a compact (nine point) stencil, which is a good compromise between efficiency
and complexity. We aim at exploring finite difference schemes with wider stencils and develop a general
stability theory in the future.
The plan of the article is as follows. Section 2is devoted to the definition of the finite difference scheme
in the whole space and to the stability analysis by means of the energy method without any boundary.
Previous stability results for this numerical scheme, such as in the references [LW64,Tad86,Cou14], were
relying on the Fourier transform, which is not convenient for half-space or quarter-space problems. In
Section 2, we recover the optimal stability criterion for the Lax-Wendroff scheme (2.2) below by only using
elementary energy arguments (discrete integration by parts and Cauchy-Schwarz inequalities). As far as
we know, even this part of our analysis is new. We then apply a similar strategy in Section 3to deal with
half-space problems. Our main result in this section is Theorem 3.1 in which we prove that the first order
extrapolation boundary condition (see (3.2) below) maintains stability for the corresponding numerical
scheme in a half-space. We even recover a trace estimate for the solution, which is in agreement with the
fulfillment of the Uniform Kreiss-Lopatinskii condition (see, e.g., [GKS72,GKO95,Mic83]). Eventually,
we consider in Section 4.1 the quarter-space with a transport operator that is outgoing with respect
to both sides of the boundary. In view of Theorem 3.1, we may expect that enforcing an extrapolation
numerical boundary condition on each side is a good starting point for deriving a stable scheme. However,
the extrapolation procedure on each side of the boundary still leaves one undetermined quantity at each
time step, which is the value of the numerical solution at the corner of the space domain. Applying
our energy argument, we are able to propose an extrapolation numerical corner condition that maintains
stability. Numerical evidence suggests that “wrong” corner conditions may yield a strongly unstable
scheme.
2 Stability for the Cauchy problem
2.1 Definition of the numerical scheme
We consider the two-dimensional transport equation on the whole space R2:
(∂tu+a ∂xu+b ∂yu= 0 , t ≥0,(x, y)∈R2,
u|t=0 =u0,(2.1)
where a, b are some given real numbers. We make no sign assumption on a, b in this section. The initial
condition u0in (2.1) belongs to the Lebesgue space L2(R2;R). We consider below a finite difference
approximation of (2.1) that is defined as follows. Given some space steps ∆x, ∆y > 0 in each spatial
2
direction, and given a time step ∆t > 0, we introduce the ratios λ:= ∆t/∆xand µ:= ∆t/∆y. In
all what follows, the ratios λand µare assumed to be fixed, meaning that they are given a priori of
the computation and are meant to be tuned in order to satisfy some stability requirements (the so-
called Courant-Friedrichs-Lewy condition [CFL28], later on referred to as the CFL condition, see for
instance Corollary 2.4 below). The solution uto (2.1) is then approximated on the time-space domain
[n∆t, (n+ 1) ∆t)×[(j−1/2) ∆x, (j+ 1/2) ∆x)×[(k−1/2) ∆y, (k+ 1/2) ∆y) by a real number un
j,k for
any n∈Nand (j, k)∈Z2. The discrete initial condition u0is defined for instance by taking the piecewise
constant projection of u0in (2.1) on each cell, that is (see [GKO95]):
∀(j, k)∈Z2, u0
j,k := 1
∆x∆yZ(j+1/2) ∆x
(j−1/2) ∆xZ(k+1/2) ∆y
(k−1/2) ∆y
u0(x, y) dxdy .
This initial condition satisfies:
∆x∆yku0k2
`2(Z2)=X
(j,k)∈Z2
∆x∆y(u0
j,k)2≤ ku0k2
L2(R2).
It then remains to determine the un
j,k’s inductively with respect to n. The Lax-Wendroff scheme with a
stabilizer reads (see [LW64]):
un+1
j,k =un
j,k −λ a
2un
j+1,k −un
j−1,k−µ b
2un
j,k+1 −un
j,k−1
+(λ a)2
2un
j+1,k −2un
j,k +un
j−1,k+(µ b)2
2un
j,k+1 −2un
j,k +un
j,k−1
+λ a µ b
4un
j+1,k+1 −un
j+1,k−1−un
j−1,k+1 +un
j−1,k−1(2.2)
−(λ a)2+ (µ b)2
8un
j+1,k+1 −2un
j+1,k +un
j+1,k−1
−2un
j,k+1 + 4 un
j,k −2un
j,k−1+un
j−1,k+1 −2un
j−1,k +un
j−1,k−1,
where (j, k) belongs to Z2. The so-called stabilizing term corresponds to the last two lines on the right
hand side of (2.2). This term is meant to add some (rather weak) dissipation that improves the stability
properties of the finite difference approximation. We refer to [LW64,GKO95] for alternative approxima-
tions of (2.1).
We first recall some stability results for the numerical scheme (2.2) and then propose an energy method
in order to recover the optimal stability criterion for (2.2). The relevance of the energy method is made
more precise below in Sections 3and 4.1 when we extend our approach to more involved geometries.
2.2 A reminder on the Fourier approach
The `2stability of the iteration (2.2) was first analyzed in [LW64] (for symmetric hyperbolic systems) by
means of the numerical radius of the amplification matrix. When one specifies the result of [LW64] to the
scalar case, the main result of [LW64] shows that (2.2) is stable in `2(Z2;R), that is:
∀n∈N,kun+1 k`2(Z2)≤ kunk`2(Z2),
if, and only if, the parameters λ, µ satisfy the restriction:
(λ a)2+ (µ b)2≤1
2.(2.3)
3
The extension of this result to symmetric hyperbolic systems is the purpose of [LW64,Tad86,Cou14].
However, all these references are based on Fourier analysis and a sharp estimate of the numerical radius
or of the norm of the amplification matrix. This technique is of little use for half-space or quarter-space
problems as we intend to study below. We thus propose below an alternative proof of the stability result
of [LW64] for (2.2) by using the energy method. We restrict for simplicity to the scalar case since our
main concern is to deal with extrapolation procedures for outgoing transport equations. Our goal is to
recover the same sufficient condition (2.3) for stability of (2.2) in `2(Z2;R) (the necessity of (2.3) for
stability is proved in [LW64] by computing the amplification matrix at the frequency (π, π)).
2.3 The energy method
We now explain how the energy method gives the optimal stability criterion (2.3) for (2.2) on the whole
space Z2. Our main result is Corollary 2.4 at the end of this section. Since the result is not new, we rather
focus on the method and all intermediate steps and postpone the statement of the main result to the end
once all preliminary steps have been achieved. We thus start from the definition (2.2) and decompose
un+1
j,k into three pieces:
∀(j, k)∈Z2, un+1
j,k =un
j,k −wn
j,k +vn
j,k ,
where vn
j,k and wn
j,k are defined by:
vn
j,k := −λ a
2un
j+1,k −un
j−1,k−µ b
2un
j,k+1 −un
j,k−1,(2.4a)
wn
j,k := −(λ a)2
2un
j+1,k −2un
j,k +un
j−1,k−(µ b)2
2un
j,k+1 −2un
j,k +un
j,k−1
−λ µ a b
4un
j+1,k+1 −un
j+1,k−1−un
j−1,k+1 +un
j−1,k−1(2.4b)
+(λ a)2+ (µ b)2
8un
j+1,k+1 −2un
j+1,k +un
j+1,k−1
−2un
j,k+1 + 4 un
j,k −2un
j,k−1+un
j−1,k+1 −2un
j−1,k +un
j−1,k−1,
It will be useful below to use operator notations in order to highlight symmetry or skew-symmetry
properties. We thus introduce the following discrete first order partial derivatives and Laplacians:
(D1,+U)j,k := Uj+1,k −Uj,k ,(D1,−U)j,k := Uj,k −Uj−1,k ,
(D2,+U)j,k := Uj,k+1 −Uj,k ,(D2,−U)j,k := Uj,k −Uj,k−1,
D1,0:= D1,++D1,−
2, D2,0:= D2,++D2,−
2,∆1:= D1,+D1,−,∆2:= D2,+D2,−.
In order to keep the notation as simple as possible, we write below D1,+uj,k rather than (D1,+u)j,k and
analogously for other operators. We hope that this does not create any confusion. With such definitions,
the operators D1,0and D2,0are skew-selfadjoint on `2(Z2;R) and the operators ∆1, ∆2are selfadjoint on
`2(Z2;R), see [GKO95] (this is known as discrete integration by parts or Abel’s transform). We also have
D∗
1,−=−D1,+and D∗
2,−=−D2,+for the h·;·i`2(Z2)scalar product. Let us eventually observe that all
operators defined above commute, which will also be useful below.
The above definitions allow us to rewrite (2.4) in a compact form as:
vn:= −λ a D1,0un−µ b D2,0un,(2.5a)
wn:= −(λ a)2
2∆1un−(µ b)2
2∆2un−λµabD1,0D2,0un+(λ a)2+ (µ b)2
8∆1∆2un.(2.5b)
4
As a consequence, we observe that vngathers all the skew-selfadjoint operators acting on unand wn
gathers all the selfadjoint operators acting on un. In particular, we easily obtain the following relation:
kun+1 k2
`2(Z2)− kunk2
`2(Z2)=kwnk2
`2(Z2)+kvnk2
`2(Z2)−2hun;wni`2(Z2),(2.6)
because vnis orthogonal to both unand wnin `2(Z2;R). The stability analysis of the Lax-Wendroff
scheme (2.2) in the whole space Z2then relies on the following two results, whose proof will be given
below.
Lemma 2.1. Let un∈`2(Z2;R), and let the sequences vn, wn∈`2(Z2;R)be defined by (2.5). Then there
holds:
kvnk2
`2(Z2)−2hun;wni`2(Z2)=−(λ a)2
4k∆1unk2
`2(Z2)−(µ b)2
4k∆2unk2
`2(Z2)
−(λ a)2+ (µ b)2
4kD1,+D2,+unk2
`2(Z2).
Proposition 2.2. Let un∈`2(Z2;R), and let the sequence wn∈`2(Z2;R)be defined by (2.5b). Then
there holds:
4kwnk2
`2(Z2)≤2(λ a)2+ (µ b)2n(λ a)2k∆1unk2
`2(Z2)+ (µ b)2k∆2unk2
`2(Z2)
+(λ a)2+ (µ b)2kD1,+D2,+unk2
`2(Z2)o.(2.7)
The remainder of this section is devoted to the proof of Lemma 2.1 and Proposition 2.2. In the end,
we explain how these two results give a stability, and even a dissipativity, estimate for the Lax-Wendroff
scheme (2.2) under suitable CFL conditions. Before proving Lemma 2.1 and Proposition 2.2, we state a
first crucial lemma which will be very useful below and will also guide us in the analysis of the half-space
and quarter-space problems.
Lemma 2.3. Let U∈`2(Z2;R). Then there holds:
kD1,0Uk2
`2(Z2)=kD1,+Uk2
`2(Z2)−1
4k∆1Uk2
`2(Z2),(2.8a)
kD2,0Uk2
`2(Z2)=kD2,+Uk2
`2(Z2)−1
4k∆2Uk2
`2(Z2),(2.8b)
kD1,0D2,0Uk2
`2(Z2)=kD1,+D2,+Uk2
`2(Z2)
−1
4kD1,+∆2Uk2
`2(Z2)−1
4kD2,+∆1Uk2
`2(Z2)+1
16 k∆1∆2Uk2
`2(Z2).(2.8c)
Proof of Lemma 2.3.Let us start with (2.8a) and (2.8b). Given three real numbers Uj−1, Uj, Uj+1, there
holds the relation:
(Uj+1 −Uj−1)2
4+(Uj+1 −2Uj+Uj−1)2
4=1
2(Uj+1 −Uj)2+1
2(Uj−Uj−1)2.(2.9)
We now observe that for a square integrable sequence U∈`2(Z;R), the two terms on the right hand side
of (2.9) have equal sum: X
j∈Z
(Uj+1 −Uj)2=X
j∈Z
(Uj−Uj−1)2.
5
摘要：

TheNeumannboundaryconditionforthetwo-dimensionalLax-Wendro schemeAntoineBenoit*&Jean-FrancoisCoulombel„October12,2022AbstractWestudythestabilityofthetwo-dimensionalLax-Wendro schemewithastabilizerthatapprox-imatessolutionstothetransportequation.Theproblemis rstanalyzedinthewholespaceinordertoshowth...

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