An error estimate for finite volume methods for multidimensional conservation laws

Abstract

In this paper, an L°°(Ll)-error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained. These methods can be formally high-order accurate and are defined on general triangulations. The error is proven to be of order ft'/4 , where h represents the "size" of the mesh, via an extension of Kuznetsov approximation theory for which no estimate of the total variation and of the modulus of continuity in time are needed. The result is new even for the finite volume method constructed from monotone numerical flux functions.

Kruzkov’s estimates for scalar conservation laws revisited

“…On the other hand the final result, a convergence rate in h 1/4 rather than h 1/2 in one space dimension (see R. Sanders [17]), is very easy to explain. This loss just comes from the BV estimate, which blows like h −1/2 (while it is bounded in one dimension); see S. Champier, T. Gallouët, R. Herbin [3], B. Cockburn, F. Coquel, P. Le Floch [4][5], A. Szepessy [18], J.-P. Vila [19], B. Cockburn, P.-A. Gremaud [7], R. Eymard, T. Gallouët, R. Herbin [9].…”

“…with η (ξ) = ξf (ξ) (see [3], [18], [4], [5], [19], [7], [9] and §5). To deal with a general φ, the inequalities (1.7) are not sufficient since the entropy dissipation is neglected.…”

Finite volume relaxation schemes for multidimensional conservation laws

“…Error estimates of difference schemes to relaxation models arising in chromatography were proved in [ScTW], [ShTW]. The convergence of finite volume schemes approximating the entropy solution of (1.3) was analyzed, e.g., in [CCL1,2], [KR], [V]. In a recent paper Rohde [R], using an appropriate extension of DiPerna's theory, has proved convergence of finite volume schemes to weakly coupled hyperbolic systems.…”

“…Error estimates of order O(h 1/4 ) for finite volume approximations to (1.3) were previously obtained in [CCL1], [V], and for finite elements in [CG1]. For finite difference approximations the order of convergence O(h 1/2 ) was established, e.g.…”

“…The main reason for the reduced order of convergence in the finite volume case is the lack of BV bounds for the approximate schemes in unstructured meshes. To compensate this, an estimate for the discrete gradients in L 2 was proved in [CCL1], [V], which led to the O(h 1/4 ) estimate. In the case of relaxation schemes considered here we are able to prove an analogous bound, cf.…”