4 Properties of the discrete solution

Error estimate for the approximation of non-linear conservation laws on bounded domains by the nite volume method

Mario Ohlberger Julien Vovelle y

Abstract

In this paper we derivea priorianda posteriorierror estimates for cell centered nite volume ap- proximations of non-linear conservation laws on polygonal bounded domains. Numerical experiments show the applicability of thea posterioriresult for the derivation of local adaptive solution strategies.

Keywords: hyperbolic equation, initial-boundary value problem, nite volume method, error estimate

MSC: 35L65,65N15

1 Introduction

Let be an open convex polygonal bounded domain in IR^d,^d= 2 3, endowed with the Euclidean norm

j j and let ^{T 2} IR⁺. We consider the following initial boundary value problem for non-linear scalar conservation laws:

ct+^r F(^{x t c}) = 0 in (0 ^T) (1)

c( 0) = ^c0 in (2)

c(^{x t}) = ^c(^{t x}) in^@(0 ^T)^: (3) The ux in equation (1) is given by the functionF^2C1((0 ^T)IRIR^d) the functions^c02L1() and^c2L1(^@(0 ^T)) are respectively the initial and boundary data of the problem (1){(3).

The nite volume methods are known to be well-suited for the discretization of conservation laws. A basic account for this claim is the fact that, by construction, they respect the conservation principle which constitutes the root of equation (1). Indeed, the evolution of the discrete unknown ^cK in each control volume^K is given by the equation

jKjcnK⁺¹=^jKjc_nK+ ^{tn X}

2@K

Qn (4)

in which we denote abusively by^@Kthe set of faces of^Kand where^jKjis the volume of^K. Equation (4) is the expression of the fact that the discrete evolution of ^cK is governed by the values of the discrete uxes^Qnacross the boundary of^Kin the time interval ^{tn tn+1}]. It is the choice of this numerical uxes

Qn that determine the nite volume method. In what follows, we will specically consider three-point nite volume schemes with monotone uxes (see (13){(17)). This category of schemes encloses all relevant rst order three-point nite volume schemes.

Since both (1) and (4) are evolution equations, the main features of the analysis of conservation laws, and of their approximations by the nite volume method already appear in the context of the Cauchy problem, i.e. = IR^d and no boundary conditions have to be taken into account. The order of accuracy of the nite volume method for the Cauchy problem is one of these well-known features: the rst given

Abteilung fur Angewandte Mathematik, Universitat Freiburg, Hermann-Herder-Str.10, D-79104 Freiburg, mario@mathematik.uni-freibrug.de

yUniversite de Provence, CMI, F-13453 Marseille, vovelle@cmi.univ-mrs.fr

a priorierror estimate is the (sharp) ^h1=2(^hbeing the size of the mesh) estimate of Kuznetsov Kuz76]

in the 1D case. This estimate remains valid on structured meshes in IR^d while, for nite volume schemes on unstructured meshes, the lack of an uniform^BV estimate on the numerical solution leads to an error estimate of reduced order ^h1=4 KNR95, BCV95, CCL95a, EGH00]. Still, in the context of the Cauchy problem, rened error estimates have been given (and their sharpness analyzed) according to the genuine non-linearity of the ux, to the structure of the entropy solution to (1)-(2), or to the nature of the waves in the solution. We refer to the discussion and compilation made by T. Tang on that profuse subject Tan01].

For practical applications a posteriori error estimates are even more important than just convergence rates. Such estimates allow to extract error indicator information that can be used in order to derive ecient self adaptive strategies for the nite volume schemes. A posteriori error estimates for nite volume approximations to the Cauchy problem were rst derived by Tadmor Tad91] in one space dimension, and by Cockburn and Gau CG95] in the multi dimensional case. A localized estimate for general ux functions and the derivation of self adaptive schemes was given in KO00]. Further results for nite volume approximations to the Cauchy problem were obtained in GM00, KO03, KKP02], while nite element approximations to the Cauchy problem were studied in JS95, SH95, HMSW99, HH02]. We emphasize that up to now no a posteriori results are available for approximations of the initial boundary value problem (1){(3).

Although the study of the nite volume method applied to the Cauchy problem (1), (2) has led to the understanding of most of the mechanisms which govern the accuracy of this numerical method of approximation, the initial-boundary value problem (1){(3) has its own interest (for the simple and major reason that the domains under consideration in practical applications can be bounded domains), and its approximation by nite volume schemes deserves an analysis. With that purpose in mind, notice that a new and characteristic feature of the approximation of the initial-boundary value problem (1){(3) by a nite volume scheme is the possible creation of a numerical boundary layer. This numerical boundary layer is a sub-product of the numerical diusion eects induced by the scheme. Of course, its presence is also related to the way in which the boundary data are implemented in the scheme. Let us specify this point. We consider here and in the following the implementation of boundary data via "ghosts control volumes". This is a way to compute the numerical uxes at the boundary of the domain inspired by the design of the uxes inside the domain. Indeed, if is an edge of a control volume^K but also one of the edges of the control volume ^L then the numerical ux ^Qn (cf. (4)) is given as a function of the discrete unknowns ^c_nK and ^c_nL by the formula ^Qn =^Gn(^c_nK ^c_nL) where, among other properties, the function^Gn is non-decreasing with respect to its rst argument and non-increasing with respect to the second. If, edge of a control volume^K, is now located at the boundary of the domain , then a ghost control volume^Lsuch that^LIR^dn and =^K\Lis introduced and the computation of the numerical ux at the boundary^Qnrelies on a numerical ux function^Gn(non-decreasing with respect to its rst argument and non-increasing with respect to the second) via the formula^Qn=^Gn(^c_nK ^c_nL) where the value ^c_nL is a discretization (typically the mean value) of the boundary datum ^c on ^{tn tn+1}). This method of computation of the numerical uxes at the boundary of the domain is classical and ensures the convergence of the nite volume scheme to the entropy solution of the problem (1){(3) Sze91, BCV95, CCL95a, Vov02]. Let us also stress that the proposed nite volume discretization is of rather importance for practical applications (see the discussion on the implementation of numerical boundary conditions in the approximation of two-phase ow problems in EGV03]). Before coming back to our considerations on numerical boundary layers, and on their inuence on the speed of convergence of the nite volume method, let us observe that, whensystemsof conservation laws are considered, the computation of the numerical uxes at the boundary of the domain by the method of ghost control volumes may be not accurate. Other methods, like reecting, or absorbing boundary conditions are in use, and, when used, the method of ghost control volumes is associated to the Godunov method for the computation of the ux. In this context, the Godunov method is indeed considered to give the reliable choice of numerical ux functions at the boundary.

The study of the numerical boundary layer has been performed by C. Chainais-Hillairet and E. Grenier CHG01], in the 1D case and for modied Lax-Friedrichs schemes on cartesian grids in the multi-D case.

Such an analysis gives a precise description of the numerical solution and, as a consequence, the speed of convergence of this solution to the entropy solution of the problem (1){(3). In the non-characteristic case

with smooth exact solutions, this speed of convergence is proved to be of order^hin the^L1(0 ^TL1()) norm, where^his the size of the mesh.

Unfortunately, the techniques of numerical boundary layer analysis seem dicult to be set when no selected direction of (discrete) derivation exists, as is the case when nite volume schemes on unstructured meshes are used. For such schemes one can therefore think to adapt the technique developed by Kuznetsov Kuz76] for the analysis of the Cauchy problem in the framework of the initial boundary value problem to get error estimates, with the drawback that this tool is not accurate at all to take into account the special phenomena at the boundary of the domain. In the specic situationF(^{x t c}) =u(^{x t})^f(^c) with

f monotone, this drawback can be overcome, for the reason that the inow and outow parts of the boundary are determineda prioriby the given velocity eldu. In Vig97], Vignal gives an a priori error estimate of order ^h1=4 for the initial boundary value problem. However, to our knowledge, for general uxesF, and general schemes on possibly unstructured meshes no results or techniques of error estimates which account for the inuence of the boundary condition have been delivered. In order to ll in this gap, we adapt the technique of Kuznetsov Kuz76] to the proof of uniqueness of the entropy solution given by F. Otto Ott96, MNRR96], and prove that the error can be estimated by ana posteriori error bound which is at least of order^h1=6 for meshes with mesh size^h. (see Propositions 5.1, 5.2). The order,^h1=6, of oura priorierror estimate has also to be discussed. Our comments are postponed to Remark 5.9.

Since the nite volume methods introduce some numerical diusion eects in the approximation of the entropy solution to the problem (1){(3), they are often related to the approximation by the vanishing viscosity method with, say, a viscosity of (small) order^". In IV03, DIV03] are developed the tools (notion of kinetic solution for the initial boundary value problem) and given the proof of an error estimate of order^"1=3.

The article is structured as follows. In Section 2 are given and recalled some properties of the entropy solution to the problem (1){(3). In Section 3 are dened the nite volume schemes under consideration some of their properties are explained in Section 4 while in Section 5 are proved the error estimates which are the center of our study. Finally, in Section 6 we give numerical experiments to illustrate our analysis. We complete the presentation with the proof of a BV estimate on the entropy solution on convex polygonal bounded domains in Appendix A.

2 Properties and regularity of the exact solution

Problem (1){(3) for general ux F, and in the context of entropy solutions has rst been analyzed by C. Bardos, A.-Y. LeRoux and J.-C. Nedelec BLN79] in the ^BV framework. The notion of entropy solution given by the three authors has been extended, in the^L1 setting, by F. Otto Ott96, MNRR96].

We present and use this last denition, by using the following semi Kruzhkov entropy-entropy ux pairs Ser96, Car99].

Notation 2.1 (Semi Kruzhkov entropy-entropy ux pairs). Let^a>b(resp. ^a?b) denote the max- imum (resp. the minimum) of ^aand ^b, set ^s+=^s>0,^s;= (^;s)⁺ and denote by sgn(^s)the derivative of the function^s+ (resp. ^s;) with the value0at^s= 0. We denote by(^s )the entropy ux associated to the entropy(^s;), that is to say

(^{x t s} ) =sgn(^s;)(F(^{x t s})^;F(^{x t} ))^:

We will often drop the dependence of over the variables^xand^tand shorten the notation to(^s ).

Notation 2.2. We denote by^Cm and ^CM²IRsome lower and upper bounds for the data:

Cm^{c0 cC}M a.e.

set^C= max(^jCm^{j jC}M^j)and let^Lbe a xed real satisfying

L max^fjFc(^{x t c})^j (^{x t c})²(0 ^T)^C_m ^C_M]^g: (5)

Denition 2.3 (Entropy solution). A function^c2L1((0 ^T))is called an entropy weak solution of (1){(3), if it satises the following entropy inequalities: for all^2Cm ^CM], for all^'2C_c¹(IR^dIR⁺) with^' 0,

Z

(0T⁾(^c;)^@t^'+ (^c ) ^r'+^Z

(^c0;)^'( 0) +^LZ_@

(0T⁾(^c;)^' 0^: (6) The space^L1is preserved by equation (1), as well as the space ^BV, and we have the following theorem.

Theorem 2.4 (Existence, uniqueness, regularity).

Let^c02L1(), ^c2L1(^@(0 ^T)). Suppose that F^2C1((0 ^T)IR)and divxF(^{x t c}) = 0 for all(^{x t c})²(0 ^T)IR. Then there exists a unique entropy weak solution^c2L1((0 ^T))of the problem (1){(3) which is bounded by the data as follows

jjc(^t)^jj_L¹⁽⁽⁰_T⁾⁾max^fjjc0jj_L^{1() jjcjj}_L¹⁽_@⁽⁰_T^))g:

If furthermore, ^{c0 2 BV}(), and ^{c 2 BV}(^@(0 ^T)), then ^{c 2 BV}((0 ^T)) and there exists a constant^CBV ^>0which depends on the data and on only such that

jjcjjBV⁽⁽⁰T^))CBV^: (7)

Proof. We refer to BLN79, Ott96, MNRR96, Vov02] for the results of existence and uniqueness of the entropy solution. In BLN79] is given a^BV estimate on the entropy solution, which requires to be^C2. The^BV estimate in the case where is a polygonal bounded domain is a new result and we give the rather involved and technical proof in Appendix A.

Remark 2.5 (BLN). Under the hypotheses of Theorem 2.4, let ^c be the entropy solution of the prob- lem (1){(3). Suppose that^c2BV((0 ^T))and denote by^c the trace of the function^con^@(0 ^T). Then:

1. ^c satises the following entropy inequalities: for all ^2C_m ^C_M]:

Z

IR +

(^c;)^@t^'+ (^c )^r'+^Z

(^c0;)^'( 0)^;Z_@

(0T⁾(^c ) n^' 0^: (8) for all^'2C_c¹(IR^dIR⁺)with^' 0.

2. Moreover, the so-called BLN condition BLN79] is satised by^con the boundary of the domain: for a.e. (^{x t})^2@(0 ^T), for allin the interval with extremities^c(^{x t}) and ^c(^{x t}):

(^c ) n 0^: (9)

3. The inequality (8), together with (9), implies (6). Indeed, if(^{x t})^2@(0 ^T)and if^c(^{x t}) then (9) gives

;^;(^c(^{x t}) ) n0 =^L(^c(^{x t})^;)^; while, if ^>c(^{x t})then (9) gives^;(^{c c}) n 0and

;^;(^c(^{x t}) ) n(^;(^{c c})^;;(^c )) n^L(^c(^{x t})^;)^;:

4. In fact, it is possible to prove MNRR96, Vov02] that for every function^wwhich is measurable and bounded a.e. on ^@(0 ^T), one has

;(^c(^{x t}) ^w(^{x t})) n^L(^c(^{x t})^;w(^{x t})) (10) for a.e. (^{x t})^2@(0 ^T).

3 Notations, assumptions and the denition of the scheme

In this section we will x the notations and assumptions and dene the nite volume scheme for solving (1){(3).

Assumption 3.1. The data of problem (1){(3) are supposed to satisfy the following conditions:

c

0 2 L

1

\BV(IR^d) ^c2L1\BV(^@(0 ^T))

F ^{2 C1}((0 ^T)IR) divxF(^{x t c}) = 0 ⁸(^{x t c})²(0 ^T)IR^: (11) The initial and boundary data are supposed to belong to the space^L1\BV. This makes sense if one has in mind practical applications in which these data are physical or biological quantities. The hypotheses of regularity and divergence-free on the ux^F are also in coherence with the possible physical or biological underlying model for equation (1). The divergence free condition in (11) may be removed, and source terms may be considered in equation (1) as well.

Let us now give the description of the meshes and schemes used to solve (1){(3). Let^J :=^{ft0 ::: t}N^gbe a partition of 0 ^T] and ^tn:=^tn+1;tnbe the step size of^J. For each^n2f0 ^{::: Ng}let^Tn=^fTj^jj2In

int

be a regular triangulation of . The joint edge of^Tj and ^Tl will be denoted by^Sjl. g

The set of internal edges^Sintn and the oriented set of internal edges^Eintn are assimilated to the sets of the corresponding indexes and are respectively dened as

Sn

int:=^f(^{j l})^{2Iintn Iintn gjS}jl is an interior edge of^{Tng Eintn} :=^f(^{j l})^{2Sintn jj>l g:}

As mentioned in the introduction, we use the concept of ghost cells to compute the ux at the boundary.

We therefore introduce the notations related to the use of this method. Let the index set^Iextn be such that

In

ext

\In

int =and such that for each edge^S@ there exists a unique pair of indices (^{j l})^{2Iintn Iextn} with ^@Tj ^\S = ^S. In this situation we denote ^Sjl := ^S. Accordingly, the set of edges located on the boundary of is denoted by

Sn

ext :=^f(^{j l})^{2Iintn Iextn jS}jl is an exterior edge of^Tng:

We also denote by^h_nmin:= minj²Iⁿdiam (^Tj) the size of the mesh at time^tn. The mesh^Tnsatises the following structural hypothesis:

Assumption 3.2. There exists a real ^>0such that for all^hj :=diam(^Tj),^j2I:

hdj ^jTj^{j j@T}j^{j h}dj^;1: (12) In order to design the nite volume approximation, we rst dene the class of monotone numerical uxes in use.

Denition 3.3 (Numerical uxes). The numerical uxes are functions ^g_njl^2C(IR² IR), each for any (^{j l}) ^{2 Sn} and ^tn ² IR⁺, satisfying the following conditions (respectively: monotony, convervativity, regularity, consistency).

8v 2^Cm ^CM] ^gnjl(^v )is monotone non-increasing on ^Cm ^CM]

8w2^Cm ^CM] ^g_njl( ^w)is monotone non-decreasing on^Cm ^CM] (13)

8v w2^C_m ^C_M] ⁸(^{j l})^{2Sintn g}_njl(^{v w}) =^;g_nlj(^{w v}) (14)

8w v w 0

v 0

2^Cm ^CM] ^jg_njl(^{v w})^;g_njl(^{v0 w0})^jLjSjl^j(^jw;w0j+^jv;v0j) (15) and

gnjl(^{w w}) = 1^t

Z tⁿ⁺¹ tⁿ

Z

S^{j l}F(^{x t w}) ⁿjl dx dt (16)

whereⁿjl denotes the outer unit normal to^Sjl with respect to^Tj.

Denition 3.4 (Finite volume scheme). Set

c 0j := 1

jTj^j

Z

T^{jc0 j2I}

0

int

cnl:= 1^t 1

jSjl^j

Z tⁿ⁺¹ tⁿ

Z

S^jlc(^{x t})dx dt (^{j l})^{2Sextn :} The discrete evolution of the approximate value^cj of ^c in the cell^Tj is governed by the equation

cnj⁺¹:= ^c_nj^{; t}

jTj^j

X

l²N⁽j⁾

gnjl(^c_nj ^c_nl) ^{j 2Iintn} (17) for all^n2f0 ^{::: Ng}, where^N(^j) denotes the index set of the neighboring cells of^Tj including the ghost indices across the boundaries of the domain.

Given the discrete values ^c_nj, we denote by^ch the approximate solution^ch: (0 ^T)^!IRdened by

ch(^{x t}) :=^cnj if ^x2Tj ^tnt<tn⁺¹: (18) The stability of the explicit scheme (17) is ensured under the following CFL condition.

Assumption 3.5 (CFL - condition). We assume the following CFL-condition, for a given ²(0 1): ^tn (1^;) ^2h_nmin

L :

4 Properties of the discrete solution

As the entropy solution of problem (1){(3), the discrete solution is ^L1 stable. On the contrary, the validity of ^BV estimates on ^ch is still an open question (in the case where unstructured meshes are considered): only \weak^BV estimates" are known. These two aspects of the behavior of the discrete solution are detailed in the following two lemmas (see Vov02] for a proof).

Lemma 4.1 (L¹ - stability). Let^chbe the discrete solution (17) and let the Assumptions 3.1, 3.2 and 3.5 be fullled. Then the function ^ch satisfy the following^L1 estimates:

jjch^jjL¹⁽⁰T^])max^fjjc0jj_L^{1() jjcjj}_L¹⁽_@⁽⁰_T^))g and

Cm^cnj^CM for all(^Tj ^tn)^2TnJ:

Lemma 4.2 (Weak BV estimate). Let^ch be the discrete solution (17) and let the Assumptions 3.1, 3.2 and 3.5 be fullled. Then there exists^C 0only depending on ^{c0 c L T} and such that

X

tⁿ²J

X

(jl^{)2Eintn tn}( max_cⁿ

l

abc^nj(^g_njl(^{b a})^;g_njl(^{b b})) + max_cⁿ

l

abc^nj(^g_njl(^{b a})^;g_njl(^{a a})) ^pC

h

(19) and

X

tⁿ²J

X

j²Iⁿ

jTj^jjcnj^+1;cnj^{j pC}

h

: (20)

Entropy inequality satis ed by the approximate solution

In Section 2 we recalled that problem (1) has a unique weak solution conforming to the entropy inequality (6). In this subsection we will show that the approximate solution^chfullls an analog inequality, including a small error term. To comparediscretetocontinuousequations, let us introduce the following forms^E+ and^E_h⁺:

Denition 4.3. The discrete function^ch being dened by (3.4) and ²IR, we set

E

h+(^ch ^') :=^;X

tⁿ²J

X

j²Iⁿ

(^cn_j^+1;)^+;(^c_nj^;)^{+ tn}

Z tⁿ⁺¹ tⁿ

Z

T^j'(^{x t})dx dt

; X

tⁿ²J

X

j²Iⁿ

1

jTj^j

Z tⁿ⁺¹ tⁿ

Z

T^j'(^{x t})^X

l²N⁽j⁾

gnjl(^c_nj^{> c}_nl^>)^;g_njl( )dxdt

E

+(^ch ^') :=^Z

IR +

(^ch(^{x t})^;)^+@t^'(^{x t}) dx dt

+ ^Z

IR +

⁺(^ch(^{x t}) ) ^r'(^{x t})dx dt +

Z

(^c0(^x)^;)^+'(^x 0) dx+^L

Z

@⁽⁰T⁾(^c(^{x t})^;)^+'d(^x) dt for any ^'2C1(0 ^T)).

The discrete (and local) entropy inequality given in Lemma 4.4 is the main account for the approximate continuous entropy inequality detailed in Lemma 4.5.

Lemma 4.4 (Discrete entropy inequality). Let ^ch be the discrete solution dened in 3.4 and let Assumptions 3.1, 3.2 and 3.5 be fullled. Then we have

E

h+(^ch ^') 0^: (21)

Proof. The discrete entropy inequality (21) follows from the monotonicity properties of the numerical uxes. See,e.g. Vov02].

Lemma 4.5 (Continuous entropy estimate). Let^chbe the discrete solution dened in Denition 3.4 and let the Assumptions 3.1, 3.2 and 3.5 be fullled. Then we have

E

+(^ch ^') ^;X

tⁿ²J

X

j²Iⁿ

jcnj^+1;cnj^jZT^j

Z tⁿ⁺¹

t^{n j'}t(^{x t})^jdx dt^;

Z

jch(^x 0)^;c0(^x)^j'(^x 0)dx

; X

tⁿ²J

X

(jl^)2Eintn 2 max_cⁿ

l

abc^nj(^g_njl(^{b a})^;g_njl(^{b b}))^h_njl ^jr'j+^j't^ji

; X

tⁿ²J

X

(jl^)2Eintn 2 max_cⁿ

l

abc^nj(^g_njl(^{b a})^;g_njl(^{a a}))^h_nlj ^jr'j+^j't^ji

; X

tⁿ²J

X

(jl^)2Eintn

L(^jc_nj^j+^jc_nl^j)^h_njl ^jr'j+^j't^ji

;L Z

@⁽⁰T⁾(^ch^;c)^+'(^{x t})^d(^x)dt^{; X}

tⁿ²J

X

(jl^)2Sextn

L(^cl+^cj^;2^Cm)^h_nlj ^jr'ji(22) where the Radon measures _njl,_njl,_njl are dened as

hnjl ^{g i} := ^hj+ ^{tn tnjT}j^jjSjl^j

Z tⁿ⁺¹ tⁿ

Z

T^j

Z tⁿ⁺¹ tⁿ

Z

S^{j l}

Z

1

0

g(+^#(^x;) ^s+^#(^t;s)) d^#d ds dx dt

hnjl ^{g i} := (^hj+ ^tn)^{2 tnjS}jl^j

Z tⁿ⁺¹ tⁿ

Z

S^{j l}

Z tⁿ⁺¹ tⁿ

Z

S^{j l}

Z

1

0

g(+^#(^;) +^#(^s;))d^#d dt d d

hnjl ^{g i} := ^hj

jTj^j

Z tⁿ⁺¹ tⁿ

Z

T^j

Z

S^{j l}

Z

1

0

g(+^#(^x;) ^t) d^# d dx dt^:

Here, we have introduced the discrete boundary datum^chdened by^ch^jS^{j l}tⁿtⁿ⁺¹⁾:=^c_nlfor(^{j l})^2Sextn .