Rate of Convergence of a Particle Method for the Solution of a ... - Inria

Rate of Convergence of a Particle Method for the Solution of a 1D Viscous Scalar Conservation Law in a Bounded Interval Author(s): Mireille Bossy and Benjamin Jourdain Source: The Annals of Probability, Vol. 30, No. 4 (Oct., 2002), pp. 1797-1832 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/1558835 Accessed: 26/12/2009 17:14 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ims. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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TheAnnals of Probability 2002, Vol. 30, No. 4, 1797-1832

RATE OF CONVERGENCE OF A PARTICLE METHOD FOR THE SOLUTION OF A 1D VISCOUS SCALAR CONSERVATION LAW IN A BOUNDED INTERVAL BY MIREILLE BOSSY AND BENJAMIN JOURDAIN

INRIAand ENPC-CERMICS In this paper, we give a probabilisticinterpretationof a viscous scalar conservation law in a bounded interval thanks to a nonlinear martingale problem. The underlying nonlinear stochastic process is reflected at the boundary to take into account the Dirichlet conditions. After proving uniqueness for the martingale problem, we show existence thanks to a propagationof chaos result. Indeed we exhibit a system of N interacting particles, the empirical measure of which converges to the unique solution of the martingaleproblem as N --- +oo. As a consequence, the solution of the viscous conservationlaw can be approximatedthanksto a numerical algorithmbased on the simulationof the particle system. When this system is discretized in time thanks to the Euler-Lepingle scheme [D. Lepingle, Math. Comput.Simulation 38 (1995) 119-126], we show that the rate of convergenceof the erroris in O(At + 1/v/,), where At denotes the time step. Finally,we give numericalresults which confirmthis theoreticalrate.

1. Introduction. We are interested in the following viscous scalar conservation law with nonhomogeneous Dirichlet boundary conditions on the interval [0, 1]:

a

-v(t,

at

x) =-

v(1 )v( (,

v(t, O) =

'2 a2

2 x2 v(t, vo(x)

Vx

x)-

a

A(v(t

ax

E [0,

x))

(t, ) E (0, +0)

x (0, 1),

1],

and v(t, 1) = 1,

Vt > 0.

We suppose that A :R -- R is a C1 function and that the initial data vo is the cumulative distributionfunction of a probability measure Uo on [0, 1], which is written, Vx E [0, 1], vo(x) = Uo([0, x]) = H * Uo(x), where H(y)= 1(y>0} denotes the Heaviside function. After giving a probabilistic interpretationof the solution of this equation thanksto a nonlinearmartingaleproblem,we want to derive and study a particle approximationof this solution. Our main motivationis that the spatial domain in (1) is bounded.To our knowledge, the only paperabout a probabilisticparticle interpretationfor the solution of a partialdifferentialequationposed in a bounded Received January2001; revised December2001. AMS2000 subjectclassifications.65N12, 65C35, 60K35, 60F99. Keywordsand phrases. Reflected stochastic processes, nonlinear martingale problem, Euler discretizationscheme, weak convergencerate. 1797

1798

M. BOSSY AND B. JOURDAIN

spatial domain is [1], which is dedicatedto the two-dimensionalincompressible Navier-Stokes equation. In [1], the authorsdo not prove the convergenceof the proposedparticlemethod. By consideringthe much simplerequation(1), we are able not only to prove the convergencebut also to boundthe associatedrate. When the viscous scalar conservationlaw is posed in the spatial domain R insteadof [0, 1], one can show thatits uniqueweak solutionis equal to H * Pt (x), where (Pt)t>o denote the time-marginals of the probability measure P on C([0, +00), IR)characterizedthe following martingaleproblem nonlinearin the sense of McKean [4, 8]: (i) P = Uo, (ii) V

?E Cb2(R),

p(Xt) - (p(Xo)-

t0 'r^2A -p (Xs) + A'(H * Ps(Xs))p' (Xs) ds 2

is a P-martingale,where X denotes the canonicalprocess on C([0, oo), IR). Here, we follow a similar approach. To take into account the Dirichlet boundary conditions, we work with a diffusion process with reflection. That is why we introduce (X, K), the canonical process on the sample path space C = C([0, +oo), [0, 1]) x C([0, +oo), IR)(endowedwith the topology of uniform convergence on compact sets). For P in ' (C) the set of probabilitymeasures on (, (Pt)t>o is the set of time-marginalsof the probability measure P on C([0, +oo), [0, 1]) definedby P = P o X-1. We associatethe following nonlinear problemwith (1). A probability measure P E P(Ce) solves the martingale at problem(MP) starting Uo 0 So e 3 ([O,1] x R), if the following hold: DEFINITION 1.1.

(i) P o (X, K)-

0o;

=Uo

(ii) V 0 E C2 (R),

qp(Xt Kt) - qp(Xo- Ko) -t

(2

s - Ks) + A'(H * Ps(Xs))p'(X

- Ks) ds

is a P-martingale; (iii) P-a.s., Vt > O, fodlKls fo(l -2Xs)dlKIs.

< +oo, IKlt = flj{o,(Xs)dlKIs jo

and Kt =

The finite variationprocess Kt which increases when Xt = 0 and decreases when Xt = 1 accounts for reflection and prevents Xt from leaving the interval [0, 1]. In Section 2, we prove that if P solves problem (MP), then (t, x) -- H * Pt (x) is

the uniqueweak solutionof (1). We deduceuniquenessfor the martingaleproblem.

1799

PARTICLEMETHODFOR A CONSERVATIONLAW

Existenceis obtainedthanksto a propagationof chaos resultfor a system of weakly interactingdiffusionprocesses. In Section 3, we discretizethis system in time thanksto the version of the Euler scheme introducedby Lepingle [10]. This way, we derive a numericalmethod to approximatethe solution of (1). We prove a theoreticalrate of convergence in ((At + 1/V/N), where At and N denote respectively the time step and the numberof particles.This rate is the same as the one obtainedby Bossy [3] when the spatial domain is R. As an importantstep in the proof, we show that, when applied to a one-dimensionaldiffusion reflected at the boundaryof [0, 1] with nondegenerateconstantdiffusioncoefficient,the weak errorof the Euler-Lepingle scheme is in (9(At). To our knowledge,this is the firstresultconcerningthe weak errorof this scheme. The last section is devoted to numerical experiments which confirm the theoretical rate of convergence of our particle method. The treatmentof the reflectionby the Euler-Lepingle scheme does not alter the convergencewhereas we exhibit a sublinearnumericaldependenceon the time step At when the particle system is discretizedthanksto the cruderEulerprojectionscheme. To conclude the Introduction,we should mentionthat, using signed weights as in [8] and [3], we could extend our approachto deal with the following more general boundary conditions in (1): Vt > 0, v(t, O) = a and v(t, 1) = b, and

Vx E [0, 1], vo(x) = Uo([0, x]), where Uo is a boundedsigned measureon [0, 1] satisfying the compatibilitycondition Uo([0, 1]) = b - a. However, we restrict ourselves to a simple case without weights to avoid furthercomplication of the alreadytechnicaldevelopments. 2. Probabilistic interpretation of the viscous scalar conservation law equation.

For T > 0 let QT = (0, T) x (0, 1) and let W2 ((2),

W2'(T)

denote the Hilbertspaces with respectivescalarproducts(cf. [9]) (u,V),W2

,

T)

=

(U V)W21'1(T) =

(uv + x (uv

xv)dxdt,

+ axu axV + atu atv)dxdt.

We introducethe Banachspace V2 (02T) = {u e W2' ((T) such that

C((, T), L2(0, 1))

= < IlullvO1,(Tr) supo 0, Pt has a density pt which belongs to L2([0, 1]) and it holds that lpt IIL2([0,1])< C(1 + t-1/4)exp(Ct).

We still have to check that V satisfies the identity (2). Let 0 be a C?? function on [0, T] x [0, 1] with 0(t, 0) = 0(t, 1) = 0 for all t E [0, T]. We set /t(t, x) = fo ?(t, y)dy. According to Definition 1. (ii), under the probabilitymeasure P, (Xt - Kt - fo A'(V(s, Xs)) ds) is a local martingalewith quadraticvariationt, that is, a Brownianmotion. By Definition 1.1(iii) and the choice of the boundary conditions for 0, P-a.s.,

4t

o ax

Xd(s, Xs).dK

O

(s, {oX,

l(Xs)dKs=0.

Thus, by Ito's formula

t E(t,

2

a22(X

X0) + E Jo s + 2 ax2s t + a lf + E -t- (s, Xs)A'(V(s, Xs)) ds.

Xt) = E(0,

O Ax

X)

1803

PARTICLEMETHODFOR A CONSERVATIONLAW

= av (s *), we deduce that

As

V??-av fo'1 )(t, (t, x) dx = x)

avo aavx dx

0(O S,-(x)

ax ao

1

V

ft +

| |

OOa+SIax

2-

(sx'as)-

(s,x)dxds

7(s,x) {(S,x) d

1 ax , oav [t +| x)A'(V(s,x))-(s, 10((s, o'o'ax

x ds x) dx ds.

Applying the Stieltjes integrationby parts formula in the spatial integralsin the firstand last lines of the previousequality,we get identity(2) for 0 a C?? function vanishing for x = 0 and x = 1. As V is in V?'1(M2T)we can extend the identity

easily by density for any function0 in I 2' ((T). Hence V(t, x) = H * Pt(x) is a weak solution in the sense of Definition2.1. Uniqueness for the martingaleproblem (MP) is derived from the uniqueness resultfor the problem(1): if P and Q solve (MP), then, for any (t, x) E [0, +00) x R, H * Pt (x) = H * Qt(x). Hence P and Q solve a linearmartingaleproblemwith boundeddriftterm A'(H * Pt(x)) and by the GirsanovtheoremP = Q. D PROOFOF LEMMA 2.4.

We just have to adapt to the case of reflected diffu-

sion processes the proof of Proposition1.1 of [12]. Accordingto Definition 1.1(ii), underthe probabilitymeasureP, (Xt - Kt- fOA'(V(s, X)) ds) is a Brownian motion. As suP[o1]IA'(x) < +oo, by the Girsanovtheorem,underthe probability measure Q e 3 (e) such that dpQ = , where ft cxp i 1

Zt = expyf

1

-

A'(H * Ps(Xs))d(Xs - Ks) - 2-- A

H

-

\

P(Xs))ds)

(Xt - Kt)

is a Brownianmotion startingat aXo and (Xt)t>o is the doubly reflectedprocess associatedwith (apt)t>O. For 4f bounded and measurable,since EPe(r(Xt)) = E]Q(Vr(Xt)Zt), by the Cauchy-Schwarzinequality, /3t =

E^P(!(Xt))


+oo.

PROOF. Except in the treatmentof the discontinuityof the Heavisidefunction, we follow the proof given by Sznitman[13], Theorem1.4. Whenpossible, we take advantageof the particularform of our diffusion domain (the interval [0, 1]) to simplify the arguments. By Proposition 2.3, uniqueness holds for problem (MP). As the particles (Xi'N, Ki'N)i 1, t > s > sl > *. > sp > 0, g E Cb(R2p) and define a mapping F on the set Y((C) of probabilitymeasures on e by F(Q) = (Q, g(Xs , Ks, ... Xsp, Ksp) x (((Xt-Kt)-p(Xs

-

-Ks)

t(Xr - Kr)+A'(H*

Qr(Xr))'(Xr

-

Kr)dr))

The mapping Fk defined like F with the Heaviside function H replaced by the Lipschitz continuous approximation Hk(x) = k(x + -)lI{-1/k 2, the couple (XIN, X2 N) has a density that belongs to L2([0, 1] x [0, 1]) with a norm smallerthan C(1 + r-1/2) exp(Ct). Hence Vr E [0, t],

E((Hk

- H)(X'N

-

X2,N)) < C(t)(l + r-1/2)k-1/4

By (7), we deduce that lim supklimsupN E7NIF - Fk(Q)I = 0. Hence each term of the right-handside of (6) is nil andE7 IF( Q)I = 0. As a consequence,Jr -a.s., Q satisfies condition (ii) in Definition 1.1.

1806

M. BOSSY AND B. JOURDAIN

Let us check condition(iii). Since the following subset of C, FTM = (x, k): IkIT < M, IkIT= T{o,}(xs)dlkIs Vt < T, kt =

L'

(1-2xs)dlkls

is closed as stated in Lemma 2.6, {Q E P(C(): Q(FTM) > 1 - } is also closed. By the weak convergenceof 7rN to 7r?, we deduce rr?({Q: Q(FTM)

> 1 - E) > limsup rN({Q: Q(FTM)

> 1-

})

N

= (8)

1-

Q: Q({IkIT > M}) > e})

liminfrN({ N

N/(QIkIT)\

> 1 - liminfErN (Q, IklT) N

ME

liminfN E(IK N IT) Me As IK'NIT = f (1 - 2Xl'N)dKl'N, |K 1oIT=

by Ito's formula, we obtain 2,N - (1,N

X(1,N--

+

(2Xs1N -1)A'(H,

+

oa(2X,N-

2

1\

)2 *

N(Xl,N)) ds

1) dWl +2T.

Hence supN E(I K 1NIT) < +00. With (8), we deduce that 7r?({Q: Q(UM>O FMT) > 1 - e})

1. As E is

arbitrary,we conclude that 0 7rZ

QQ \

U FM,T) =lh T>OM>0

i

=1,

thatis, r??-a.s., Q satisfies Definition l.l(iii), which completes the proof. D LEMMA2.6. The subset FT,M of C which consists in the couples (x, k) such that IkIT < M, IklT = fo 1{o,l(xs)dIkI and Vt 0, Ikn T < M, by extraction of a subsequence, we can suppose that the measures d Ikln(resp. dkn)

on the compact set [0, T] converge weakly to da, a positive measure with

1807

PARTICLEMETHODFOR A CONSERVATIONLAW

mass smaller than M (resp. db a signed measure). As kn converges uniformly to k on [0, T], t E [0, T] -> kt is the cumulative distributionfunction of the measure db. If f: [0, T] - IRis continuous, as Xn converges uniformly to x on [0, T], =

f(s)dbs

lim

= lim

f(s)dkn

f(s)(1 - 2x)dlkn

f (s)(1 - 2xs)da,.

Hence (1 - 2xs) is a density of db w.r.t. da and Vt E [0, T],

(9)

kt =

(1 - 2x)das.

As (xn, k) E FTM, OT xs (1 - xs) dlknls = 0. Lettingn -- +oo, we get fj Xs(1- xs) das = 0, thatis, das a.e. xs E {0, 1} and 11 2x I= 1. With(9), we deducethatda is the total variationof dk andconclude that (x, k) FT,M.

O

COROLLARY 2.7. It is possible to approximatethe weak solution V(t, x) = H * Pt(x) of (1) thanks to the empirical cumulative distributionfunction H * tlN(x) of the particle system.Moreprecisely (t, x) E [0, +oo) x [0, 1],

lim

N-->+oo

EIV(t,x)

- H

* tN(x)| t

=0.

PROOF. For t > 0 and x E [0, 1], according to Lemma 2.4, the function Q E ?P(C)-- IH * Pt(x) - H * Qt (x) E IR is continuous at P. The weak convergence of the sequence (7 N)N to 7r? = 8p implies

lim

N-* + ox EIH

* Pt(x) - H * fN(x)I = E

H * P t(x)-

t(x)j = 0.

In the case t = 0, we conclude with the strong law of large numbers.

Zl

3. Particle method. In this section we describe a numerical particle method to approximate the solution V of (1) on [0, T] x [0, 1] (where T is a positive constant) and analyze its rate of convergence. According to Corollary 2.7, it is

possible to approximateV(t, x) by the empiricalcumulativedistributionfunction H * liN(x) = N >1l H(x - XjN) of the particle system (5). To transform this convergence result into a numerical approximationprocedure,we need to discretize in time the N-dimensional stochastic differentialequation (5). To do so we use the version of the Euler scheme introducedby Lepingle [10] which mimics the reflection at the boundary.We choose At > 0 and L E N such that T = L At and denote by Yt the position of the ith particle (1 < i < N) at the

1808

M. BOSSY AND B. JOURDAIN

discretizationtime tl = 1At (0 al. When at the end of the time step the computedposition is smaller than 0 or greaterthan 1, then Yti is respectivelychosen equal to 0 or equal to 1. We will actuallylet ao andai dependon At to reducethe computationaleffort,but to simplify notationwe do not emphasizethis dependenceunless necessary.Taking advantageof the one-dimensionalspace domain, we invertthe initial cumulative distributionfunction Vo(x) = H * Uo(x) to constructthe set of initial positions of the numericalparticles: yO= inf z: H * Uo(z) -> N

(10)

for 1 < i a,}

sup (Yt -+1 sE[tl ,t]

r(W - Wtil) (s

-

tl)A(V(tl,t))).

Since it is possible to simulatejointly the Brownianincrement(Wtl - Wtl) and the corresponding

supE[t ,t+l](Ws

- Wt + (s - tl)a), this discretization scheme

is feasible. To obtainthe optimalrateof convergenceO( 1/ N + At) we aregoing to make rather strong assumptionson the initial condition vo(x) = H * Uo(x) ensuring that the weak solution of (1) is in fact a classical solution. These hypotheses are possibly too restrictivebut they avoid furthercomplications of the already technical proof. For the solution of (1) to be classical, that is, C1,2 (C1 in the time variable t and C2 in the space variable x), it is necessary that vo is C2. Moreover,since the Dirichletboundaryconditions are constantin time, for x = 0 2 or x = 1, 2-a3xxv(t,x) - A'(v(t, x)) axv(t, x) = atv(t, x) = O. At time t = 0, = 2A'(0)v'(0) and we obtain the necessary compatibility conditions a2v(0) 0 00)=2/()0vlai

1809

PARTICLE METHOD FOR A CONSERVATION LAW

a2vo(1) = 2A'(1)vO(1).Ourhypothesis vo E C2+t([O, 1])

(v0'Holdercontinuouswith exponent P,), where / e]0, 1[, (H) A is a C3 function, a2v(0) =2A'(0)vo(0) and a2v(1) = 2A'(l)vo(l), is slightly strongerthan these necessary conditions. CombiningTheorem 6.1 of [9, pages 452-453], which gives existence of a classical solutionon [0, T] x [0, 1] for (1), and the proof of Lemma2.2, which gives uniquenessof weak solutionson [0, T] x [0, 1], we obtainthe following lemma. LEMMA 3.1. Under hypothesis (H), the solution V(t, x) = H * Pt(x) of (1) belongs to C1'2([0, T] x [0, 1]) and a V(t, x) is Holder continuouswith exponent

(1 + P)/2 in the time variable t on [0, T] x [0, 1]. To reduce the effort needed to compute the correctionterms C' in (11), it is interestingto let ao and al dependon the time step At and convergerespectively to 0 and 1 as At -* 0. Supposing that these convergencesare not too quick, we obtainthe following estimatefor the convergencerate of the particlemethod. THEOREM 3.2. Under hypothesis (H), if we assume that 0 < ao(At) < al (At) < 1 satisfy ao(At) A (1 - al(At)) > a AtY for 0 < y < 1/2 and a > 0,

then there exists a strictlypositive constant C dependingon A, Uo, T, a, a and y such that VO xxy}(>

( X) b(

is a continuousprocess with boundedvariation,

s, XY)) ds +

lx>>y}(dKX

- dK ).

The third term of the right-hand side is nonpositive. By the Lipschitz continuity of x -+ b(s, x) and Gronwall's lemma, we deduce that, for some real constant CT, (14)

a.s.,

sup (Xt - XY)+ < CT(X - y)+ tE[O,T]

the symmetric inequality for (XY - Xf)+, we obtain that a.s. - X I < CTIx - y L. According to the KolmogorovcontinuitythesuPtE[O, T] IXt the orem, C([0, T], [0, 1])-valued process (t -- XY) indexed by y e [0, 1] admits a continuous version that we still denote by Xy to simplify notation. By (14), a.s., Vq < q' E [0, 1] n Q, Vt [0, T], Xq < Xq'. With the a.s. continuity of Using

PARTICLE LAW1 METHOD FORA CONSERVATION

1811

y e[0, 1] -+ X?' in the variable y, we conclude that this function is a.s. nondecreasing. Now let Xo be an initial variable with law Uo independent of the Brownian motion W. We easily check that the law of (XtTOKf O)tEro,T] on C([0, T], [0, 1] x IR) solves the following martingale problem:

(i)Qo = Uo ( 60; (ii) ~p(Xt- Kt) - qp4Xo- Ko) - ft 02 p"(X - Ks) ? A'(V(s, Xs))yp'(XsKs) ds is a Q martingale for any function y E C (R); (iii) Q a.s., Vt > 0, fodtIKI < +oo, IKit = f0to,i},j(Xs)dIKIs and Kt = By the Girsanov theorem, uniqueness holds for this martingale problem. Since the image of the solution P of problem (MP) starting at Uo 0 so by the canonical restriction from C([0, +oo), [0, 1] x IR) to C([0, TI, [0, 1] x IR) solves this martingale problem, we deduce that V (t, x) E [0, T] x [0, 1], V(t, x) XO ,~~ D~Tj~ff, H * Pt~J (x) = E(H (x - X )). By independence of Xo and W, we conclude that V(t, x) = E(j1 H(x - XY)Uo(dy)). D

We easily deduce thatthe initializationerroris smallerthan 1/N. LEMMA3.4.

For all

t > 0, N

sup V(t,x) xE[O,1]

PROOF. Let Uo= a.s. ,

jk =f16i

N __

Clearly, -

H(x -XbUo(dy) =

y

?LIE(H(x-Xto))

WLH(x

- xv)

(Uo - Uo)({Y E [0, 1]:XY < XJ).

Since, by Lemma 3.3, y YX is a.s. continuous and increasing, if nonempty, the < XY set {y: x} is equal to [0, (Pt(x)], where /t(x) = inf{y : X > x}. By definition of the initial positions y6 [see (10)], Vy E [0, 1], 0 < (Uo - Uo)([O, y]) < I/N. Hence

0


0, L E N) and tl = At for 0 < 1 < L. The Euler-Lepingle discretization of the stochasticdifferentialequation t Xt = y + a Wt +

b(s, XY) ds + K,

t KI dlls, 10,Ii}(Xy)

(15)

IKYlt=

-

(1- 2Xy)diKy s

is given by X-y = Y, Xto XY =Ov (XY +a(Wt Ct = {xa

t

sup ( ~-

+a (Ws - Wtl)+-b(tl,X)(s

-t))+

sE[tl,t]

Vt E [tl, t/+l]. coefficient the drift condition on In the next proposition, assuming a regularity b(s, x) which is satisfied by A'(V(s, x)) under hypothesis (H) (see Lemma 3.1), we upper-bound the weak convergence rate of this scheme: PROPOSITION3.5. Assume that b is C1'2 on [0, T] x [0, 1], that for some a > O, axb(t, x) is Holder continuous with exponent a in t and that 0 < ao(At) < al(At) < 1 satisfy ao(At) A (1 - al(At)) > a AtY for y e [0, 1/2) and a > 0. Then there is a constant C depending on a, T, b, a, y but not on y and At such

that, when f: [0, 1] -> R is a function with boundedvariation and m denotes its distribution derivative, Vl < L,

IE(f(XY) - f(XtY)) I< C At

ml(dx).

The error proceeds from two sources. The first one is the usual Euler discretization of the drift coefficient. The second contribution is the inexact treatment of the reflection on the lower boundary (resp. the upper boundary, resp. andtl+l when XtY> al [resp. Xti < ao, resp. Xl e both boundaries) tlbetween (ao, al)], which will turn out to be negligible. To get rid of it, we introduce

the Euler-Peano discretizationof (15). The Euler-Peano scheme is a theoretical discretization scheme which consists in freezing the drift coefficient on each interval[tl, tl+l] whereasthe normalreflectionremainsexact: y + aWt + b( rs, XY) d + Kt, t t ( K16)e= f{ o, I1(XY)dlKY ls, (1KtY= Xt

(16)

where Tr= At[ s ] and [x] denotes the integralpartof x.

2XY)d KY Is,

1813

PARTICLE METHOD FOR A CONSERVATION LAW

Assume that b(., ) is bounded and that 0 < ao(At)
a AtY for y E [0, 1/2) and a > 0.

Then,for some positive constantsc and C independentof At and y, (17)

V1 < L

P(3k (aA tY- sup lb(, )t) AO / ).

Since y < 1, for At small enough, a Aty - sup lb(., ?)I At > a AtY/2. Then ZT Y i ,y
a Aty and remarking that Vt E [tk, tk+l], Ct < r supsE[tk,tk+l] IW - WtkI+ sup Ib( , )I A t, we easily obtain similar bounds for

I(X = Xt a, Xk+ Xtk+,)and P(Xtk conclude with (18) since 1 < L = T/At. [-

Xtk

ao, Xtk+l: X

and we

Let 1 > 1, let f: [0, 1] -> IRbe a function with bounded variationand let m denote its distributionderivative.Accordingto the previouslemma, |E(ft(XY)-

f(XY))|

|< IE(f (XY)f +

sup

(xt )) If(z)--

f(z')lCAt-Y-1/2e-

At2y-1

z,z'e[O, 1]