Finite-volume WENO schemes for three-dimensional conservation laws

In this paper we firstly carry out an extension of the finite-volume WENO schemes to three space dimensions and higher orders of accuracy. Secondly, we propose to use more accurate fluxes as the building block. These are the HLLC and MUSTA fluxes [Multi-stage predictor–corrector fluxes for hyperbolic equations, 2003; Restoration of the contact surface in the HLL Riemann solver, Report CoA 9204, June 1992; J. Shock Waves 4 (1994) 25]. The numerical results suggest that the new WENO-HLLC and WENO-MUSTA schemes compare satisfactorily with the state-of-the-art finite-volume scheme of Shi et al. [J. Comput. Phys. 175 (2002) 108].