A weighted average flux method for hyperbolic conservation laws

Abstract

A numerical technique, called a ‘weighted average flux’ (WAF) method, for the solution of initial-value problems for hyperbolic conservation laws is presented. The intercell fluxes are defined by a weighted average through the complete structure of the solution of the relevant Riemann problem. The aim in this definition is the achievement of higher accuracy without the need for solving ‘generalized’ Riemann problems or adding an anti-diffusive term to a given first-order upwind method. Second-order accuracy is proved for a model equation in one space dimension; for nonlinear systems second-order accuracy is supported by numerical evidence. An oscillation-free formulation of the method is easily constructed for a model equation. Applications of the modified technique to scalar equations and nonlinear systems gives results of a quality comparable with those obtained by existing good high resolution methods. An advantage of the present method is its simplicity. It also has the potential for efficiency, because it is well suited to the use of approximations in the solution of the associated Riemann problem. Application of WAF to multidimensional problems is illustrated by the treatment using dimensional splitting of a simple model problem in two dimensions.