A novel finite volume scheme for hyperbolic conservation laws

Abstract

AbstractIt is well known that the conventional shock capturing schemes for hyperbolic systems of conservation laws yield oscillatory solutions near discontinuities when the mesh is not fine enough. In this article, an improved finite volume scheme (IFVS) is proposed to simulate hyperbolic conservation laws. In this scheme, some weighting coefficients are introduced to discretize the convection term. By eliminating the truncation error of the numerical scheme, the weighting coefficient and the ratio of space step to time step are determined analytically. Theoretical and numerical results show that although the same computational stencil as the classical finite volume method (FVM) is employed to divide the computational domain and the simple explicit forward difference is used to discretize the time derivative, the proposed IFVS can achieve almost the same accuracy as the exact solution for the scalar hyperbolic conservation laws. Furthermore, for the hyperbolic conservation laws involving shocks, the numerical solutions obtained by the IFVS do not show any unphysical oscillations even a very coarse mesh is utilized to divide the computational domain, which is consistent with the theoretical analysis showing that the IFVS is unconditionally stable.