Increasingly high-order ALW-MR-WENO schemes for solving hyperbolic conservation laws on tetrahedral grids

Abstract

This paper proposes a new fourth-order multi-resolution weighted essentially non-oscillatory scheme (which is termed as the MR-WENO-4 scheme), and the new MR-WENO-3 and MR-WENO-4 schemes with adaptive linear weights (which are termed as the ALW-MR-WENO-3 and ALW-MR-WENO-4 schemes) in the finite volume framework for solving hyperbolic conservation laws on tetrahedral grids. It is the first time to devise three increasingly high-order WENO schemes on tetrahedral grids, since one MR-WENO-4 scheme uses the information defined on four unequal-sized central stencils, and the ALW-MR-WENO-3 and ALW-MR-WENO-4 schemes use the information defined on two unequal-sized central stencils. In comparison to the classical third-order finite volume WENO scheme (Zhang and Shu, 2009 [53]) that used the sixteen four-cell stencils on tetrahedral grids, the key benefits of these new MR-WENO schemes are their simplicity, efficiency, and compactness in the WENO processes, the arbitrary selection of any positive linear weights without considering the topological structures of the tetrahedral grids, the ideal high-order accuracies in smooth areas, and the non-oscillatory property in the vicinity of strong discontinuities. More importantly, only two linear weights are automatically adjusted to be arbitrarily positive values on condition that one simple restriction is satisfied when designing the high-order ALW-MR-WENO schemes. Finally, some numerical results are proposed to show the good efficiency of these ALW-MR-WENO schemes which save approximately 12%-43% CPU time in comparison to that of the original MR-WENO-3 scheme and the new MR-WENO-4 scheme on tetrahedral grids.