An Error Estimate for Finite Volume Methods for Multidimensional Conservation Laws

Abstract

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.

A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations

“…In the context of nonlinear hyperbolic conservation laws these techniques where used to prove a priori estimates [22], [24], [3], [5], [8], [2] and a posteriori estimates [4], [15].…”

Finite volume schemes for nonhomogeneous scalar conservation laws: error estimate

“…It has already been done for the homogeneous conservation law (q = 0) by R. Eymard, T. Gallouët, R. Herbin [10,11], [9] (with M. Ghilani) in the case where F (x, t, s) = v(x, t)f (s) and by B. Cockburn, F. Coquel, P. LeFloch [6,7], D. Kröner, M. Rokyta [12] and J.-P. Vila [24] in the case where F (x, t, s) = F (s). In a former paper [1], C. Chainais considered the homogeneous problem u t (x, t) + div(F (x, t, u(x, t))) = 0, x∈ R N , t ∈ R + , u(x, 0) = u 0 (x), x ∈ R N (2) under the hypothesis div x F = 0.…”

“…But here, because of the source term and of the fact that div x F / = 0, the function sign plays an important role in the definition of the entropy solution (6). This function is not smooth in 0 and we may note later that the definition of u T ,k by (13) is crucial for the proof of the entropy inequalities (45) and (54).…”

Convergence of fluctuation-splitting schemes for two dimensional scalar conservation laws with a kinetic solver

“…It allows to prove that the scheme (18) is consistent with all the entropy inequalities in a meaning that will be specified further for the KFSS. Notice that no a priori compactness estimate (like TV bound) is possible and this point is common with classical finite volumes methods (see [3], [4], [5], [8], [9], [11], [12], [15], [25] for instance).…”

“…In the third section we describe the KFSS and state our convergence result. The last section is devoted to the proof of the convergence, based on a consistency argument for entropy using a weak BV estimate as in [17] in the linear case, and [3], [4], [5], [8], [9], [11], [12], [15], [25] for finite volumes schemes.…”