Convergence of an explicit finite volume scheme for first order symmetric systems

Summary.

 This paper is devoted to the derivation of a O(h ^1/2) error estimate for the classical upwind, explicit in time, finite volume scheme for linear first order symmetric systems. Such a result already existed for the corresponding implicit in time finite volume scheme, since it can be interpreted as a particular case of the space-time discontinuous Galerkin method but the technique of proof, used in that case, does not extend to explicit schemes. The general framework, recently developed to analyse the convergence rate of finite volume schemes for non linear scalar conservation laws, can not be used either, because it is not adapted for systems, even linear. In this article, we propose a new technique, which takes advantage of the linearity of the problem. The first step consists in controlling the approximation error ∥u−u _h ∥ _L2 by an expression of the form <ν _h , g>−2<μ _h , gu>, where u is the exact solution, g is a particular smooth function, and μ _h , ν _h are some linear forms depending on the approximate solution u _h . The second step consists in carefully estimating the error terms <μ _h , gu> and <ν _h , g>, by using uniform stability results for the discrete problem and regularity properties of the continuous solution.