Generalization of the finite difference method in distributions spaces

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Generalization of the finite difference method in distributions spaces

Stéphane Labbé, Emmanuel Trélat

To cite this version:

Stéphane Labbé, Emmanuel Trélat. Generalization of the finite difference method in distributions spaces. 2006. �hal-00097806�

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ccsd-00097806, version 1 - 22 Sep 2006

Generalization of the finite difference method in distributions spaces

Stéphane Labbé and Emmanuel Trélat^∗

Abstract

The aim of this article is to propose a generalization of the finite difference scheme suitable with solutions of Dirac distribution type. This type of solution is for example encountered in earthquake or explosion simulations. In such problems, the difficulty is to catch sharply a moving singular front modeled by a Dirac type distribution. We give a general framework to deal with numerical methods, and use it to build finite difference methods in distribution spaces. Numerical examples are provided for a one-dimensional wave equation.

1 Introduction

In this article, we propose a generalization of the classical finite difference method suitable with the approximation of solutions in subspaces of the Sobolev spacesH^−s,s >0. We implement the proposed method for the one-dimensional wave equation with nonregular initial data of Dirac type. A concrete situation where such a method happens to be relevant is the simulation of an earthquake or an explosion front in which an accurate approximation of the singular front of the solution is required. These objects, from a mathematical point of view, are modeled by Dirac type distributions.

In the classical finite difference method, the Dirac type distributions are usually approximated by smooth functions. This type of approximation, when injected in a temporal processus, becomes swiftly incorrect, due to the fact that the scheme is built on a regular approximation of the solution. Then, the high frequencies are ill-estimated and tend to disperse. The idea is to build a scheme dealing specifically with Dirac type distributions and their derivatives.

To illustrate the problem, consider the one-dimensional boundary value problem

u^′′(x) +c(x)u(x) =f(x), x∈(0,1),

u(0) =u(1) = 0, (1)

wherec andf are continuous functions on [0,1]. The classical finite difference scheme is the following. LetN be a positive integer, h = 1/N, and xi =ih, i= 0, . . . , N, be discretization points on [0,1]. Note that, if the functionuis of classC² on (0,1), then

u^′′(x) = u(x+h)−2u(x) +u(x−h)

h² + O(h²),

for everyx∈ (0,1). Hence, in order to solve numerically the problem (1), we are naturally led to the numerical finite difference scheme

ui+1−2ui+ui−1

h² +c(xi)ui =f(xi), i= 1, . . . , N−1, u0=uN = 0.

In order to ensure a convergence property of the classical finite difference method, a strong regularity of the solution is usually assumed. For instance, recall that, ifc(x) ≥0, for every x∈ [0,1], and if the solutionuof (1) is of class C² on (0,1), then there exists a positive real numberC, independent on N (and h), such that

0≤i≤Nmax |u(xi)−ui| ≤Ch².

In this paper, an analysis of this discretization problem suitable for nonregular solutions is achieved, leading to a numerical scheme whose convergence is proved.

The structure of this article is the following.

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In Section??, we recall the classical finite difference method. We then introduce in Section?? the concept ofdiscretization process, and provide a unified mathematical framework for projective discretization methods, ensuring convergence of the methods. We investigate the 1-D wave equation, first recovering the usual finite difference method, and then extending this method to distribution spaces. Finally, in Section??, the method is implemented and simulations are provided for nonregular initial conditions of Dirac type.

2 Discretization processes: a projective approach

2.1 Generalities

LetW andV be separable Banach spaces, andA:W →V be a bounded linear operator. Forf ∈V, we consider the problem of determiningu∈W so that

Au=f. (2)

In this section, we introduce an abstract framework in order to define rigorously a discretization process of problem (2), so as to obtain a numerical scheme of the form

Ahuh=fh, (3)

wherefh∈Vh, whereAh:Wh→Vh is a bounded linear operator representing a discretization of the operator A, in a sense to be made precise next, and whereWh andVh are suitable vector spaces approximations ofW andV. The discretization parameter h is chosen in a given nonempty open subset H of (0,+∞)^p such that 0 ∈ H, with p integer (for instance, p = 2 for the finite difference method on a time dependent problem on a space interval).

Definition 2.1. Adiscretization processis a tripleD= (W^⋆,(Wh, Ph)h∈H, W), where:

• W^⋆ andW are separable Banach spaces such thatW is a dense subset of W^⋆;

• Wh is a vector subspace ofW^⋆, for everyh∈H;

• Ph :W^⋆→Wh is a projection operator;

• ifP_h^∗ denotes the canonical injection fromWh inW^⋆, then

h→0limkP_h^∗◦Phu−ukW^⋆ = 0, for everyu∈W;

• the norms of the operatorsPhandP_h^∗ (with respect to the normsW^⋆and Wh) are bounded, uniformly with respect toh.

Remark 2.1. By definition,Ph◦P_h^∗=id_W_h andPh◦Ph=Ph.

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Wh

P_h^∗

W^⋆

P_h^∗◦Ph

Ph

W^⋆

Figure 1: Commutative diagram

The commutative diagram of Figure 2.1, for every h ∈ H, illustrates the definition.

Let (W^⋆,(Wh, Ph)h∈H, W) and (V^⋆,(Vh, Qh)h∈H, V) be two discretization processes. For everyh∈H, set

Ah=Qh◦ A^⋆◦P_h^∗,

whereA^⋆:W^⋆→V^⋆is a linear operator extendingA. Set moreoverfh=Qhf. We obtain in this way a discretized version of problem (2), of the form (3).

An important property is thewell posedness of the discretized problem (3).

A scheme is said to bewell posed if (3) admits an unique solution, for everyh.

A continuous problem will be said to be well posed ifAis bijective.

The following result is obvious.

Lemma 2.1. Assume that kerQh∩(A^∗◦P_h^∗(Wh)) = {0}, for every h ∈ H. If the extended operator A^∗ is surjective, then the discrete problem (3) has a unique solution.

Note that the well posedness of the continuous problem is not sufficient in general to ensure the well posedness of the discretized problem.

We next recall the definition of a consistent, stable, and convergent scheme.

Definition 2.2. The numerical scheme (3) is said to beconsistent with (2) if

h→0limkQ^∗_h◦ Ah◦Phu− AukV^⋆ = 0, whereuis the solution of (2).

Definition 2.3. The numerical scheme (3) is said to be stable if there exist a connected open subset H0 of H, with 0 ∈ H0, and positive constants C and ε0 such that, for everyε∈(0, ε0), every h∈ H0, and everyeh ∈Vh such that kehkVh < ε, for alluh,u˜h∈Wh satisfying

Ahuh=Qhf, Ahu˜h=Qhf+eh, there holds

kuh−u˜hkWh≤Cε.

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Assume that the problem (2) is well posed.

Definition 2.4. The numerical scheme (3) is said to be convergent if there exists a connected open subset H0 ofH, with 0∈ H0, such that the problem (3) is well posed, for everyh∈H0, and

h∈Hlim0,h→0kuh−ukW^⋆ = 0, whereuhis the solution of (3).

If there existp >0 andC >0, depending only onu, such that there holds moreover

kuh−ukW^⋆ ≤C |h|^p,

for everyh∈H0, then the numerical scheme (3) is said to beof order p.

As usually, the consistency and stability properties imply the convergence property, according to the following theorem.

Theorem 2.1. Assume that ker(Qh)∩(A^∗◦P_h^∗(Wh)) = {0}, for every h ∈ H. If the numerical scheme (3) is consistent with (2) and stable, then it is convergent.

Proof. For everyh∈H, set

eh=Qh(Q^∗_h◦ Ah◦Phu− Au),

whereuis the solution of (2). From the consistency property, threr existsC >0 such that, for everyε >0, there exists r >0 such that, for everyh∈H with

|h|< r, we have

kehkVh≤Cε,

whereC is a positive real boundingkPhkfor everyh∈H. By definition ofeh, one has

Ah(Ph(u)) =eh+Qhf.

The stability property implies that, for everyhinH0, kAhu−uhkW_h ≤CC1ε,

whereC1is the constant introduced in the Definition 2.3, anduhis the solution of (3). The hypothesis on the kernel ofPh implies, using Lemma (2.1), thatuh

is unique. Then, using the properties of the operatorP_h^∗◦Ph, there holds kP_h^∗uh−ukW^∗ = kP_h^∗uh−u+P_h^∗◦Ph(u)−P_h^∗◦Ph(u)kW^∗

≤ kP_h^∗◦Ph(u)−ukW^∗+kP_h^∗(uh−Ph(u))kW^∗

≤ kP_h^∗◦Ph(u)−ukW^∗+kP_h^∗kkuh−Ph(u)kW_h^∗

≤ kP_h^∗◦Ph(u)−ukW^∗+kP_h^∗kCC1ε.

The termkP_h^∗◦Ph(u)−ukW^∗converges to zero by hypothesis, and this concludes the proof.

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Theorem 2.2. If A^⋆ : W^⋆ → V^⋆ is boundedly invertible, then the numerical scheme (3) built on (2) converges. Let u be the solution of (2), if there exist positive constantsCu andC_u^′, andpin Ndepending ofu, such that

kQ^∗_h◦ Ah◦P_h^∗u− AukV^∗ ≤Cu|h|^p, and

kP_h^∗◦Phu−ukW^∗ ≤C_u^′|h|^p, for everyh∈H0, then the scheme is of order p.

Proof. We first prove that the numerical scheme is consistent. Let u be the solution of (2). There holds

kQ^∗_h Qh A^⋆ P^∗P u− AukV^⋆

=kA^⋆ P_h^∗ Phu+ (Q^∗_h Qh−idV^⋆)A^⋆ P_h^∗ Phu− AukV^⋆

=kA^⋆ u+Au+A^⋆(P_h^∗ Ph−idW^⋆)u−(Q^∗_h Qh−idV^⋆)A^⋆ P_h^∗ PhukV^⋆

≤ kA^⋆(P_h^∗ Ph−idW^⋆)ukV^⋆+k(Q^∗_h Qh−idV^⋆)A^⋆ P_h^∗Ph ukV^⋆

≤ kA^⋆kL(W^⋆,V^⋆)k(P_h^∗ Ph−idW^⋆)ukW^⋆+k(Q^∗_h Qh−idV^⋆)A^⋆kV^⋆

+k(Q^∗_h Qh−idV^⋆)A^⋆(P_h^∗ Ph u−u)kV^⋆.

Then, from definition 2.1, it is immediate that the discretization process is consistent.

We next prove that the discretization process is stable. Using the fact that V is dense in V^⋆ and that the union of all Q^∗_hVh for h ∈ H0 contains V we deduce that there existr, C >0 such that, for everyh∈H such that|h|< r and for everyeinVh

kA^⋆P_h^∗_n(uh−uh,e)kW^⋆ ≤CkekVh,

whereuh,e is the solution of the Equation (3) whenfh is perturbed bye. Then using the factA^⋆ is boundedly invertible, there existsC^⋆>0 such that

C^⋆kuh−uh,ekWh ≤ kA^⋆P_h^∗_n(uh−uh,e)kW^⋆ ≤CkekVh.

Hence the discretization process is stable, and using Theorem 2.1, convergent.

We next prove the second part of the theorem. Letusolution of (2) anduh

solution of (3). One has

Ahuh=Qh f, and

Au=f.

Hence

AhPhu=Qhf+Qh(Q^∗_hAhP_h^∗u− Au) Seteh=Q^∗_hAhP_h^∗u− Au. By hypothesis, there holds

kehkV^⋆ ≤Cu|h|^p.

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The stability property implies that there exists C1 > 0 such that, for every h∈H0,

kuh−PhukV^⋆ ≤C1Cu|h|^p. Then,

kP^⋆uh−ukV^⋆ = kP^⋆uh−u+P_h^∗◦Phu−P_h^∗◦PhukV^⋆

≤ kP_h^∗◦Phu−ukV^∗+kP_h^∗(uh−Phu)kV^∗

≤ C_u^′|h|^p+kP_h^∗kL(Vh,V^⋆)C1Cu|h|^p, and using Definition 2.1 we conclude.

AssumingA^⋆is boundedly invertible is quite stringent. For example, it holds for linear operators of compact inverse like the Laplacian operator provided the spaces are well chosen. However, this fact cannot be assumed in general. In the case of the finite difference scheme, in order to deal with pointwise values of functions, we need to work in aC^p space. For example, the Laplacian operator sends W1 = C²([0,1]) in W2 = C⁰([0,1]). It is then natural to define W₁^⋆ = W₂^⋆ = L²([0,1]), and thus, the definition of the extension A^⋆ becomes problematic. A solution is to build W1,h andW2,h so that it is possible to set W₁^⋆ = W1 and W₂^⋆ = W2. Then the extension of the operator A raises no problem and the convergence theorem is applicable.

In what follows, we describe a finite difference discretization of the wave equation, first onH²(I)×H¹(I), and in this case we recover the classical framework of that method, and then on (H^s(I)∩H₀¹(I))^′×(H^s+1(I)∩H₀¹(I))^′,s >0.

2.2 The case of evolution problems

It may be convenient to discretize partially a problem. Typically, for evolution problems, one may discretize the spatial variable only. In this section we focus on such evolution problems. We investigate semidiscretization processes, and total discretization processes.

Let T be a positive real number, W⁰ and V⁰ be separable Banach spaces, andP :W⁰→V⁰ be a continuous linear operator. Forf ∈V⁰, we consider the evolution problem

∂tu=Pu+f. (4)

It is a particular case of the latter section, with W = C¹(0, T;W⁰), V = C⁰(0, T;V⁰), and A = ∂t− P. The operator P extends canonically to P : W →V.

Let (W⁰^⋆,(W_h⁰, Ph)h∈H, W⁰) and (V⁰^⋆,(V_h⁰, Qh)h∈H, V⁰) be two discretization processes. SetW^⋆ = C¹(0, T;W⁰^⋆), and V^⋆ = C⁰(0, T;V⁰^⋆). For every h∈ H, set Ph = Qh◦ P^⋆◦P_h^∗, where P^⋆ : W⁰^⋆ → V⁰^⋆ is a linear operator extendingP.

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As previously, we consider the following canonical extensions:

P^⋆:W^⋆→V^⋆,

Ph:W^∗→C¹(0, T;W_h⁰),

Ph:C⁰(0, T, W⁰^⋆)→C⁰(0, T;W_h⁰), Qh:V^∗→C⁰(0, T, V_h⁰),

Ph:C¹(0, T;W_h⁰)→C⁰(0, T;V_h⁰).

We make the following assumption.

Assumption. We assume thatW⁰^⋆⊂V⁰^⋆, and that∂tW^⋆⊂V^⋆.

For the methods investigated in this paper, this assumption will be verified.

Remark 2.2. The assumption ∂tW^⋆ ⊂V^⋆ is necessary because the operator

∂tmay act as a spatial derivative, for instance in the case of the wave equation.

2.2.1 Semidiscretization processes

The aim of this paragraph is to set a precise framework in order to deal with spatial semidiscretization processes.

Setfh=Qhf. Then, seeking an approximation solution of (6) amounts to seekinguh∈C¹(0, T;W_h⁰) so that

Qh∂tP_h^∗uh=Phuh+fh. (5) Letι:W⁰^⋆ →V⁰^⋆ denote the canonical injection. It extends toι:W^⋆→ V^⋆. Then, for everyh∈H, the operators∂t andP_h^∗ commute, i.e.,

∂tP_h^∗=ιP_h^∗∂t

(see the diagram of Figure 2).

C¹(0, T;W_h⁰) C¹(0, T;W⁰^⋆)

C⁰(0, T;W_h⁰) C⁰(0, T;W⁰^⋆) C⁰(0, T;V⁰^⋆)

Ph

Ph^⋆

∂t

∂t Ph^⋆

Ph

ι

Figure 2: Diagram

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From (5), one gets

QhιP_h^∗∂tuh=Phuh+fh. Define

Mh=QhιP_h^∗

as afiltering operator. We obtain a semidiscretization scheme of the form Mh∂tuh=Phuh+fh, (6) calledsemidiscretization process.

Theorem 2.2, applied to this particular case, yields the following result.

Corollary 2.1. Given two discretization processes (W^⋆,(Wh, Ph)h>0, W) and (V^⋆,(Vh, Qh)h>0, V) and Ah, fh defined as above, under the assumption that the infinitesimal generatorP^∗ is a bounded linear operator onW^∗, the scheme (4) built on (6) converges.

Moreover, ifudenotes the solution of (2), and if there existCu, Cu^′ ≥0, and p∈Nsuch that

kQ^∗_h◦Ah◦P_h^∗u−AukV^∗ ≤Cu|h|^p, and

kP_h^∗◦Phu−ukW^∗ ≤C_u^′|h|^p, for everyh∈H, then the scheme is of order p.

Proof. This yields the existence of a nonempty open subset of H for which the scheme is stable. Furthermore, the continuity and uniform boundedness properties of operatorsPh,P_h^∗,QhandQ^∗_hensure the consistency. Theorem 2.2 implies to the convergence of the scheme.

The second part of the corollary is deduced from the boundedness of the infinitesimal generatorP.

2.2.2 Total discretization processes

A natural question is the following: is it equivalent to discretize first the spatial variable, and then the time variable, or to discretize first the time variable, and then the spatial variable?

The answer is actually positive if we deal with discretization processes, as defined previously.

Let us prove this fact sharply, and without going into details. To make short, the indexh (resp. the index k) denotes a discretization with respect to the spatial variable (resp. the time variable). Then, proving the fact amounts to proving that

Qk◦Qh◦ A^⋆◦P_h^∗◦P_k^∗=Qh◦Qk◦ A^⋆◦P_k^∗◦P_h^∗. On the one hand, it is clear that

P_h^∗◦P_k^∗=P_k^∗◦P_h^∗.

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On the other hand, sinceQhandQkare projection operators, and noticing that, up to canonical injections,Qh(ImQk)⊂(ImQk) and Qk(ImQh)⊂(ImQh), one gets

Qk◦Qh=Qh◦Qk, and the conclusion follows.

This important fact validates our approach by semidiscretization. Indeed, to make numerical simulations on a semidiscretized model, it suffices to choose a time discretization process whose order is greater than the order of the space semidiscretization process. Theorem 2.2 is used to prove the convergence of the method. Let set

V_k,h⁰ =V_h,k⁰ =Qk(C⁰(0, T;V_h⁰), W_k,h⁰ =Pk(C⁰(0, T;W_h⁰), W_h,k⁰ =Ph(Pk(C⁰(0, T;W^⋆)), then, one can announce the following corollary

Corollary 2.2.Given the discretization processes (C⁰(0, T;W^⋆),(Wh,k, PhPk)h>0,k>0, C¹(0, T;W)) and (C⁰(0, T;V^⋆),(Vh,k, QhQk)h>0,k>0, C¹(0, T;V)) and

Ah,k=Qh◦Qk◦ A^⋆◦P_k^∗◦P_h^∗,

3 Discretization of the 1-D wave equation in H

²

(I ) × H

¹

(I )

In this section, we consider the 1-D wave equation with regular data, and show how to recover the classical finite difference sheme within the abstract framework introduced previously.

3.1 Theoretical framework

Consider the Cauchy-Dirichlet problem for the one-dimensional wave equation on [0, T]×I, whereI= [a, b],

utt=uxx,

u(0,·) =u0(·), ut(0,·) =u1(·), u(t, a) =g1, u(t, b) =g2,

(7) where u0 ∈H²(I), u1 ∈ H¹(I), g1 and g2 are real numbers. It is a standard fact that this problem has a unique (weak) solution u ∈ C⁰(0, T;H²(I))∩ C¹(0, T;H¹(I))∩C²(0, T;L²(I)).

In order to point out the regularity difference betweenuandut, it is relevant to write (7) in the form

ut=v, vt=uxx,

u(0,·) =u0(·), v(0,·) =v0(·), u(t, a) =g1, u(t, b) =g2,

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Letγ1 (resp.γ2) denote the left (resp. right) trace operator onH¹(I), and letσ0 denote the trace operator onC⁰(0, T), that is,γ1u=u(a),γ2u=u(b), for everyu∈H¹(I), andσ0u=u(0), for everyu∈C⁰(0, T).

Set

W = C⁰(0, T;H²(I))∩C¹(0, T;H¹(I))∩C²(0, T;L²(I))

×C⁰(0, T;H¹(I))∩C¹(0, T;L²(I)), (9) and

V =R× C⁰(0, T;H¹(I))∩C¹(0, T;L²(I))

×C⁰(0, T;L²(I))

×R×H²(I)×H¹(I). (10) Define the operatorA:W →V by

A=







γ1 0

∂t −1

−∂xx ∂t

γ2 0 σ0 0 0 σ0





 .

Then, the previous problem is equivalent to determiningU ∈W so that

AU =





 g1

0 0 g2

u0

v0







. (11)

In what follows, set

W^∗=C⁰(0, T;L²(I))×C⁰(0, T;L²(I)), and

V^∗=R×C⁰(0, T;L²(I))×C⁰(0, T;L²(I))×R×L²(I)×L²(I).

In order to write a finite difference approximation of this operator onL²(I), we next make precise the spatial and time discretizations.

3.2 Spatial discretization

LetN be a positive integer, h= (b−a)/N, andxi=a+ih,i= 0, . . . , N, be discretization points on [a, b]. For everyi∈ {0, . . . , N}, setωi = (xi−¹₂, xi+¹₂), wherex₋¹

2 =x0,x_N₊¹

2 =xN, andx_i+¹

2 = ^xⁱ^+x₂ⁱ⁺¹, fori∈ {0, . . . , N−1}. Let χωi denote the characteristic function of the intervalωi.

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LetWf0,hdenote the set of functions onIwhose restriction to each subinterval ωi,i= 0, . . . , N, is polynomial with degree less than or equal to two.

For every positive integer m, every strictly ordered vector X ∈ R^m, and every Y ∈ Z^m, where Z is a separable Banach space, let JX(Y) denote the Lagrange interpolation polynom ofY at pointsX, that is

JX(Y)(x) = Xm

i=1

yi

Ym

j=1 j6=i

(x−xj).Y^m

j=1 j6=i

(xi−xj)

!

for everyx∈R. Note that

JX(Y)(xi) =yi, i= 1, . . . , m.

For everyi∈ {1, . . . , N−1}, setVi={i−1, i, i+ 1}, and setV0={0}, and VN ={N}.

For everyX ∈R^N⁺¹, we use the notationX.χh=PN

i=0xiχωi.

Definition 3.1. • Themean operator mh:L²(I)→R^N+1 is defined by mh(u) =

1

|ωi| Z

ωi

u(x)dx

i∈{0,...,N}

,

for everyu∈L²(I).

• Define ¯p0,h:R^N⁺¹→Wf0,hby

¯

p0,h(U) = XN i=0

J(xj)j∈Vi((Uj)j∈Vi)χωi, for everyU ∈R^N⁺¹.

• SetW0,h= ¯p0,h(R^N⁺¹) andW_0,h⁰ ={u∈W0,h |u(0) =u(1) = 0}.

• LetP0,h:L²(I)→W0,hdenote the linear mapping defined byuhk →uet P0,h(u) = ¯p0,h(M_h⁻¹mh(u)),

for everyu∈L²(I), whereMhis the (N+ 1)×(N+ 1) tridiagonal matrix

Mh=







1 0 · · · 0

1 24

11 12

1

24 ...

0 ₂₄¹ ¹¹₁₂ . ..

... . .. ... ₂₄¹ 0

1 24

11 12

1 24

0 · · · 0 1





 .

Generalization of the finite difference method in distributions spaces

HAL Id: hal-00097806

https://hal.archives-ouvertes.fr/hal-00097806

Generalization of the finite difference method in distributions spaces

Stéphane Labbé, Emmanuel Trélat

To cite this version:

ccsd-00097806, version 1 - 22 Sep 2006

Generalization of the finite difference method in distributions spaces

Contents

1 Introduction

2 Discretization processes: a projective approach

3 Discretization of the 1-D wave equation in H

(I ) × H

(I )