for Three-Dimensional Conservation Laws

Abstract

The purpose of this paper is twofold. Firstly we carry out an extension of the finite-volume WENO approach to three space dimensions and higher orders of spatial accuracy (up to eleventh order). Secondly, we propose to use three new fluxes as a building block in WENO schemes. These are the one-stage HLLC 29 and FORCE 24 fluxes and a recent multi- stage MUSTA flux 26. The numerical results in one, two and three space dimensions suggest the the new centred WENO-FORCE and up- wind WENO-HLLC and WENO-MUSTA schemes achieve a uniformly high order of accuracy for smooth solutions and produce essentially non- oscillatory profiles for discontinuities. In particular, the WENO-MUSTA scheme combines the simplicity of the centred non-staggered WENO- FORCE scheme and accuracy of the upwind WENO-HLLC scheme with a complete Riemann solver. The advantages of the WENO-MUSTA scheme will be fully realised when solving very complex hyperbolic sys- tems, such as those arising in multi-phase flows, magnetohydrodynamics and general relativity.