Convergence of a finite-volume scheme for a heat equation with a multiplicative stochastic force

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Convergence of a finite-volume scheme for a heat equation with a multiplicative stochastic force

Caroline Bauzet, Flore Nabet

To cite this version:

Caroline Bauzet, Flore Nabet. Convergence of a finite-volume scheme for a heat equation with a mul- tiplicative stochastic force. Finite Volumes for Complex Applications IX, Jun 2020, Bergen, Norway.

�hal-02442422v2�

Convergence of a finite-volume scheme for a heat equation with a multiplicative stochastic force

Caroline Bauzet, Flore Nabet

AbstractWe present here the discretization by a finite-volume scheme of a heat equation perturbed by a multiplicative noise of Itˆo type and under homogeneous Neumann boundary conditions. The idea is to adapt well-known methods in the de- terministic case for the approximation of parabolic problems to our stochastic PDE.

In this paper, we try to highlight difficulties brought by the stochastic perturbation in the adaptation of these deterministic tools.

Key words: Stochastic heat equation, Itˆo integral, multiplicative noise, Itˆo formula, predictable process, Finite Volume Method.

MSC(2010):35K05, 60H15, 65M08

1 Introduction

In this section, we present the stochastic heat equation we are studying. After deriv- ing assumptions on the data, we explain the goal of the paper and give the definition of weak solution we are looking for. More precisely, we are interested in the follow- ing stochastic heat equation set in(0,T)×Ω×Θ,

∂_t

u(t,x,ω)− Z t

0

λu(s,x,ω)dW(s)

−∆u(t,x,ω) =0, (1)

Caroline Bauzet

Aix Marseille Univ, CNRS, Centrale Marseille, LMA 13013 Marseille, France

e-mail: caroline.bauzet@univ-amu.fr Flore Nabet

CMAP, Ecole polytechnique, CNRS, I.P. Paris, 91128 Palaiseau, France e-mail: flore.nabet@polytechnique.edu

1

2 Caroline Bauzet, Flore Nabet

whereΩ is an open bounded polygonal subset ofR²,T >0,W ={W_t,Ft; 06t6 T}is a standard adapted one-dimensional continuous Brownian motion defined on the classical Wiener space(Θ,F,P)andλ ∈R. An initial condition is given by a deterministic functionu0∈L²(Ω):

u(0,x,ω) =u₀(x), x∈Ω, ω∈Θ, (2) and we consider homogeneous Neumann boundary condition :

∇u(t,x,ω)·n(x) =0,t∈(0,T),x∈∂ Ω, ω∈Θ, (3) wherendenotes the unit vector to∂ Ω, outward toΩ. In order to make the lecture more fluent, we omit in the sequel the random variable ω. Note that the present study can be easily adapted to the case whereΩ is a subset ofR³but for the sake of readability, we restrict the presentation to the 2-dimensional case.

In this paper, the stochastic integral^R₀^.λudW is understood in the sense of Itˆo, so that due to its non-anticipative construction with simple processes, we must consider an explicit time-discretization of this object. Note that the unknownuappears in the stochastic integral, thus the noise is said to be multiplicative, otherwise it is additive.

The numerical analysis of heat equation (and generally second-order parabolic equations) under stochastic perturbation, random source term or random coefficients has been the subject of several studies by the way of finite-element approximations (see [7] for a thorough presentation of the state of the art on this subject). The aim of the present paper is to expose tools of stochastic calculus used to adapt known methods in the deterministic case to get the convergence of a suitable finite-volume scheme for the approximation of problem (1)-(2)-(3). This work stands for an in- troductive study in order to apprehend more complex problems such as the finite- volume approximation of stochastic nonlinear degenerate parabolic equations, hav- ing in mind the deterministic case treated by [6].

In what follows, we will show the convergence of a suitable numerical scheme through a stochastic processu, weak solution of (1)-(2)-(3) in the following sense:

Definition 1A predictable processuwith values inL²(Ω)is a weak solution of (1)- (2)-(3) ifu∈L² (0,T)×Θ;H¹(Ω)

∩L^∞ (0,T);L²(Ω×Θ)

and if it satisfiesP- a.s inΘ and for any ψ ∈AT ={ϕ∈C^∞ R×R²

:ϕ(T, .) =0}the variational formulation

Z _T 0

Z

Ω

u(t,x)∂_tψ(t,x)dxdt− Z _T

0 Z

Ω

∇u(t,x)·∇ψ(t,x)dxdt+ Z

Ω

u0(x)ψ(0,x)dx

= Z T

0 Z

Ω Z t

0 λu(s,x)dW(s)∂_tψ(t,x)dxdt. (4)

Remark 1The predictability property of u with values in L²(Ω) is a condition of measurability of the solution u with respect to the filtration F = (Ft)_06t6T, which represents the history of the Brownian motion up to timeT. It is required since we consider a multiplicative noise. More precisely, it means thatubelongs to L² (0,T)×Θ,PT,dt⊗P;L²(Ω)

wherePT denotes the predictableσ-field gen-

erated by (see [8, p.27])

{X:(0,T)×Θ→R: Xis left-continuous and∀t∈[0,T],X_tisFt-measurable}. We denote byN_W²(0,T;L²(Ω))the space of predictable processes with values in L²(Ω). Endowed with the norm||X||²=^R₀^TR_Θ||X||²

L²(Ω)dPdt, it is a Hilbert space.

Remark 2An application of Itˆo derivation formula (see [3, Theorem 4.17, p.105]) allows us to remark that the right-hand side of (4) can also be written in the following manner

Z T 0

Z

Ω Z t

0

λu(s,x)dW(s)∂_tψ(t,x)dxdt=− Z

Ω Z T

0

λu(t,x)ψ(t,x)dW(t)dx.

2 Meshes, scheme and discrete norms

We will use a classical two-point flux approximation scheme with an admissible mesh as in [5]. Firstly, we remind for convenience this definition adapted to our sub- setΩofR²and give some notations. Secondly, we present the finite-volume scheme used to approximate the weak solutionuof (1)-(2)-(3). Thirdly, we introduce dis- creteL²(Ω)-norm andH¹(Ω)-seminorm used for the stability results exposed in the next section.

An admissible meshT is given by a family of disjoint open polygonal subsets ofΩ, called ”control volumes” and denoted byKsuch that:

• Ω =∪_K∈TK;

• ifK,L∈T,K6=L, then ˚K∩L˚=/0;

• ifK,L∈T,K6=L, either the 1-dimensional Lebesgue measure ofK∩Lis 0 or K∩Lis the edgeσof the mesh separating the control volumesKandL;

• at eachK∈T, we associate a pointx_K∈K, called the center ofK, such that if K,Lare two neighbouring control volumes, the edgeσ=K|Lwhich separatesK andLis orthogonal to the straight line going throughx_K andx_L.

Once an admissible finite-volume meshT ofΩ is fixed, we will use in the sequel the following notations.

Notations

• E[.]denotes the expectation,i.ethe integral overΘwith respect to the probability measureP.

• E is the set of the edges of the meshT andEint={σ∈E :σ6⊂∂ Ω}the set of interior edges.

• For anyK∈T,EK is the set of the edges of the control volumeKandm_K the Lebesgue measure ofK.

• For anyσ=K|L,m_σis the length ofσandd_K|Lthe distance between the centers x_kandx_L.

• h=size(T) =sup{diam(K),K∈T}, the mesh size.

4 Caroline Bauzet, Flore Nabet

In order to compute an approximation ofuon[0,T], we takeN∈N^∗and consider a fixed time step∆t=_N^T ∈R^∗+. We first define the set{u⁰_K,K∈T}by the discretiza- tion of the initial condition using its mean value over the control volumeK,

u⁰_K= 1 m_K

Z

K

u₀(x)dx. (5)

The equations satisfied by the discrete unknowns denoted byuⁿ_K,n∈ {0,· · ·,N−1}, K∈T are given by the following explicit scheme

mK

∆t(uⁿ⁺¹_K −uⁿ_K) +

∑

σ∈EK∩Eint

mσ

d_K|L(uⁿ_K−uⁿ_L) =mK

∆tλuⁿ_K(Wⁿ⁺¹−Wⁿ), (6) whereWⁿ=W(n∆t),∀n∈ {0,· · ·,N}. Note that since the Brownian motionW is standard, thus W(0) =0. The random dependence of the discrete unknownsuⁿ_K n∈ {1,· · ·,N}, comes from the randomness of the incrementsWⁿ⁺¹−Wⁿ, again for convenience, we omit the random variableωand writeuⁿ_Kinstead ofuⁿ_K(ω). We define the piecewise constant approximate solution(u^∆t_T)on(0,T)×Ω×Θ from the discrete unknownsuⁿ_Kby

u^∆t_T(t,x,ω) =uⁿ_T(x,ω) =uⁿ_K(ω) =uⁿ_K, t∈[n∆t,(n+1)∆t), x∈K, ω∈Θ, (7) where(uⁿ_T)_T defined onΩ×Θis the sequence of the approximate solution at time tⁿ=n∆tforn∈ {0, ...,N}.

Remark 3Let us mention that using properties of the Brownian motionW, for all K∈T and alln∈ {0,· · ·,N},uⁿ_KisFn∆t-measurable. Thus,u^∆t_T is predictable with values inL²(Ω)as an elementary process adapted to the filtration(Ft)_06t6T. We then define for anyn∈ {0, ..,N} the following discreteL²(Ω)-norm||.||_L2(Ω)

andH¹(Ω)seminorm|.|²_1,T for the approximate sequence(uⁿ_T)_T,P-a.s inΘ

||uⁿ_T||²_L2(Ω)=

∑

K∈T

m_K|uⁿ_K|²and|uⁿ_T|²_1,T =

∑

σ=K|L∈Eint

m_σ dK|L

|uⁿ_K−uⁿ_L|².

3 Convergence of the scheme

In this section, we propose a study of the approximate sequence(u^∆t_T). After the derivation of boundedness estimates for(u^∆t_T)independent of the discretization pa- rameters∆t andh, we propose to show the convergence of(u^∆t_T)towards a weak solutionuof (1)-(2)-(3) in the sense of Definition 1.

Proposition 1Let T >0,T be an admissible mesh, N∈N^∗ and∆t= ^T

N ∈R^∗+. Assume that the condition

∆t m_K

∑

σ∈EK∩Eint

mσ

d_K|L 61

2, ∀K∈T (8)

is satisfied. Then there exists a constant C>0only depending on T,Ω,λ and u0

such that

sup

n6N

Eh

||uⁿ_T||²_L2(Ω)

i +

N−1 n=0

∑

∆tE

|uⁿ_T|²_1,T 6C.

Proof We will principally use here properties of the Brownian motionW for the control of its discrete incrementsWⁿ⁺¹−Wⁿ(see [3, p.87]). For anyn∈ {0,· · ·,N− 1}, note that sinceE[Wⁿ⁺¹−Wⁿ] =0, one gets that for anyFn∆t-measurable ran- dom variableX:Θ→R,E[(Wⁿ⁺¹−Wⁿ)X] =E[Wⁿ⁺¹−Wⁿ]E[X] =0. Thus, by multiplying (6) byuⁿ_K and taking the expectation, sincea(b−a) =¹₂(b²−a²−(a− b)²)for anya,b∈R, one gets

m_K 2∆tE

|uⁿ⁺¹_K |²− |uⁿ_K|²− |uⁿ⁺¹_K −uⁿ_K|²

+

∑

σ∈EK∩Eint

m_σ

d_K|LE[(uⁿ_K−uⁿ_L)uⁿ_K] =0.

Moreover thanks to (6), by noting thatE[(Wⁿ⁺¹−Wⁿ)²] =∆t, one has

E

|uⁿ⁺¹_K −uⁿ_K|²

=∆tE

|λuⁿ_K|² +E





∆t

m_K

∑

σ∈EK∩Eint

m_σ

d_K|L(uⁿ_K−uⁿ_L)

!2

,

and one arrives at m_K

2 E

|uⁿ⁺¹_K |²− |uⁿ_K|²

+∆t

∑

σ∈EK∩Eint

m_σ

d_K|LE[(uⁿ_K−uⁿ_L)uⁿ_K]

6∆tm_K 2 λ²E

|uⁿ_K|² +∆t

2

∆t m_K

∑

σ∈EK∩Eint

m_σ d_K|L

!

σ∈E

∑

m_σ d_K|LE

|uⁿ_K−uⁿ_L|²

! .

Summing overK∈T and using the condition (8), thanks to classical reorderings of the summations, we obtain

K∈

∑

m_KE

|uⁿ⁺¹_K |²

+2∆t

∑

σ=K|L∈Eint

m_σ dK|L

E

|uⁿ_K−uⁿ_L|²

6(1+∆tλ²)

∑

K∈T

m_KE

|uⁿ_K|² +1

2∆t

∑

K∈T

∑

σ∈EK∩Eint

mσ

d_K|LE

|uⁿ_K−uⁿ_L|²

which leads to E

hkuⁿ⁺¹_T k²

L²(Ω)

i +∆tE

|uⁿ_T|²_1,T

6(1+∆tλ²)Eh kuⁿ_Tk²

L²(Ω)

i

. (9)

6 Caroline Bauzet, Flore Nabet

Summing (9) overn∈ {0,· · ·,m}, the discrete Gronwall lemma gives the expected L^∞(0,T;L²(Ω×Θ))bound,

Eh

ku^m_Tk²_L2(Ω)

i

6(1+∆tλ²)e^λ2T||u₀||²_L2(Ω), ∀m∈0,· · ·,N.

Summing (9) overn∈ {0,· · ·,N−1}we obtain theL² (0,T)×Θ;H¹(Ω) bound,

N−1 n=0

∑

∆tE

|uⁿ_T|²_1,T 6

1+Tλ²(1+∆tλ²)e^λ2T

||u₀||²

L²(Ω).

Remark 4Let us mention that the stochastic perturbation does not impact the con- dition (8) on the time and space discretization parameter to get a stability result on the finite-volume approximation(u^∆t_T). Indeed, the condition is the same as in the deterministic case (which corresponds toλ=0).

Remark 5Note that Proposition 1 also holds for a more general stochastic noise taking the form ^R₀^.g(u)dW withg:R→R Lipschitz-continuous. It also implies boundedness of the sequence(g(u^∆t_T))and so the existence of a weak limitg_uin NW²((0,T);L²(Ω))for a subsequence of(g(u^∆t_T)). Whengis not a linear function, the convergence result stated in Theorem 1 below requires a compactness tool com- patible with the random variable in order to affirm thatgu=g(u). This extension will be carried out in a future work.

Theorem 1For m∈N, let Tm be an admissible mesh, N_m∈N^∗ and ∆t_m = _N^T

m

satisfying the condition(8). Let(u^∆t_T^m

m)be given by(5)-(6)-(7)withT =Tmand N= Nm. Then there exists a subsequence of(u^∆t_T^m

m), still denoted(u^∆t_T^m

m), which converges weakly in L²((0,T)×Ω×Θ)towards a weak solution u of (1)-(2)-(3)in the sense of Definition 1.

Proof We will only give here the idea of the proof to handle the stochastic term and refer to [5, Proof of Theorem 18.1 p.858] for the deterministic contributions.

Letm∈N,Abe aP-measurable set andψ∈C^∞(R×R²)such thatψ(T, .) =0 and

∇ψ·n=0 on (0,T)×∂ Ω. SinceΩ is a polygonal subset ofR², the set of such functionsψ is dense in the setAT for theL²(0,T;H¹(Ω))-norm (see [4]). For the sake of simplicity we shall use the notationsT =Tm,h=size(Tm)and∆t=∆tm. We define the piecewise constant in space functionψ_T on(0,T)×Ω by

ψ_T(t,x) = 1 m_BK

Z

B_K

ψ(t,y)dy, ∀x∈K,t∈(0,T),

whereB_K⊂Kis a ball centered atx_K andm_BK denotes its Lebesgue measure. We multiply (6) by∆t1_Aψ_T(n∆t,x_K), sum the result overn∈ {0,· · ·,N−1}andK∈ T, to get after taking the expectation :T1,m+T_2,m=T3,m, where

T_1,m=E

"

1_A

N−1

∑

n=0

∑

K∈T

m_K(uⁿ⁺¹_K −uⁿ_K) 1 m_BK

Z

B_K

ψ(n∆t,x)dx

#

T2,m=E

"

1_A

N−1

∑

n=0

∆t

∑

K∈T

∑

σ∈EK∩Eint

mσ

d_K|L×(uⁿ_K−uⁿ_L) m_BK

Z

B_K

ψ(n∆t,x)dx

#

T_3,m=E

"

1_A

N−1 n=0

∑ ∑

K∈T

λm_Kuⁿ_K(Wⁿ⁺¹−Wⁿ) 1 m_BK

Z

BK

ψ(n∆t,x)dx

# .

Proposition 1 allows us to extract firstly a subsequence of(u^∆t_T), still denoted(u^∆t_T), which converges as m→+∞ to an element u in L^∞((0,T);L²(Ω×Θ)) for the weak-?topology. Secondly, since(u^∆t_T)is bounded inN_W²(0,T;L²(Ω)), one can affirm that u is predictable with values in L²(Ω) (see Remark 1) and that the previous convergence also holds (up to a subsequence) for the weak topology in N_W²(0,T;L²(Ω)). Now, following [1] (see Remark 6), [2] and [5], one shows that

T_1,m−−−−→

m→+∞ −E

1_A Z _T

0 Z

Ω

u(t,x)∂_tψ(t,x)dxdt

−E

1_A Z

Ω

u₀(x)ψ(x,0)dx

T_2,m−−−−→

m→+∞ −E

1_A Z _T

0 Z

Ω

u(t,x)∆ ψ(t,x)dxdt

.

By adapting the classical result of the two-point flux approximation scheme, one shows thatu∈L²((0,T)×Θ;H¹(Ω))by introducing a definition of discrete gradi- ent for(u^∆t_T). Since the stochastic integralI_T:X7→^R₀^TX(t,x,ω)dW(t)is linear and continuous fromN_W²(0,T;L²(Ω))toL²(Ω×Θ), it is particularly weakly continu- ous and so the regularity ofψgives

E

1_A Z

Ω Z _T

0

u^∆t_T(t,x)ψ_T(t,x)dW(t)dx

−−−−→

m→+∞ E

1_A Z

Ω Z _T

0

u(t,x)ψ(t,x)dW(t)dx

. Using successively Cauchy-Schwarz inequality onΩ×Θ, Itˆo isometry and Propo- sition 1, one shows by following [2] in the hyperbolic setting that (using the notation

|Ω|for the area ofΩ)

T_3,m−E

1_A Z

Ω Z T

0

λu^∆t_T(t,x)ψ_T(t,x)dW(t)dx

6p

|Ω|

N−1 n=0

∑ ∑

K∈T

E

"

Z

K

Z (n+1)∆t n∆t

λuⁿ_K(ψ_T(n∆t,x)−ψ_T(t,x))dW(t)

2

dx

#!¹₂

=p

|Ω|

N−1 n=0

∑ ∑

K∈T

E

Z (n+1)∆t n∆t

Z

K

λuⁿ_K(ψ_T(n∆t,x)−ψ_T(t,x))

2dxdt !¹₂

6|λ|Tp

|Ω|||∂_tψ||_∞sup

n6N

Eh

||uⁿ_T||²_L2(Ω)

i√

∆t−−−−→

m→+∞ 0.

8 Caroline Bauzet, Flore Nabet

ThusT3,m−−−−→

m→+∞ E

1_A Z

Ω Z _T

0

λu(t,x)ψ(t,x)dW(t)dx

.Finally, for anyψ∈AT

and anyP-measurable setA, one gets E

1_A

Z _T 0

Z

Ω

u(t,x)∂_tψ(t,x)dxdt

+E

1_A Z

Ω

u₀(x)ψ(0,x)dx

−E

1_A Z T

0 Z

Ω

∇u(t,x)·∇ψ(t,x)dxdt

=−E

1_A Z

Ω Z T

0 λu(t,x)ψ(t,x)dW(t)dx

,

and the result holds using Remark 2.

Remark 6In the deterministic case, Theorem 1 is classically proved by multiply- ing (6) by∆tψ(n∆t,x_K)wherex_K is the center of the control volumeK. Here, the application of Itˆo isometry gives us a coefficient √

∆t which is not sufficient to compensate the summation overn∈ {0,· · ·,N−1}. Indeed, in this case there exists C˜_ψ>0 which only depends onψsuch that

p|Ω|

N−1 n=0

∑ ∑

K∈T

E

_{Z (n+1)∆t}

n∆t Z

K

λuⁿ_K(ψ(n∆t,x_K)−ψ(t,x))

2dxdt !¹₂

6√

∆tp

|Ω||λ|

sup

n6N

Eh

||uⁿ_T||²_L2(Ω)

i¹₂ C˜_ψ

N−1

∑

n=0

(∆t+h),

but this last term does not converge to 0 when mtends to+∞under reasonable assumptions overhand∆t. Choosing here the mean-value onB_Kallows us to show both the convergence of the termsT3,mandT2,m(using forT2,msimilar arguments as in the deterministic framework, see [1, Proposition 3.5]).

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