A TVD-MUSTA scheme for hyperbolic conservation laws

Abstract

The aim of this paper is to develop an improved version of the Multi-Stage (MUSTA) approach to the construction of upwind fluxes that avoid the solution of the Riemann Problem (RP) in the conventional manner. We propose to use the second order TVD flux as a building block in the MUSTA scheme instead of the first order flux used in the original MUSTA. The numerical solution is advanced by TVD Runge-Kutta method. The new MUSTA scheme improves upon the original MUSTA and TVD schemes in terms of better convergence, higher overall accuracy, better resolution of discontinuities and find its justification when solving very complex systems for which the solution of the RP is costly or unknown. A way to extend this scheme to two dimensional systems of hyperbolic conservation laws is presented. In this paper we also extend the scheme in the framework of high order WENO methods. Numerical results suggest that our scheme is superior to the original schemes. This is specially so for long time evolution problems containing both smooth and non-smooth features.