Optimally-stable second-order accurate difference schemes for non-linear conservation laws in 3D

Abstract

In one and two spatial dimensions, Lax–Wendroff schemes provide second-order accurate optimally-stable dispersive conservation-form approximations to non-linear conservation laws. These approximations are an important ingredient in sophisticated simulation algorithms for conservation laws whose solutions are discontinuous. Straightforward generalization of these Lax–Wendroff schemes to three dimensions produces an approximation that is unconditionally unstable. However, some dimensionally-split schemes do provide second-order accurate optimally-stable approximations in 3D (and 2D), and there are sub-optimally-stable non-split Lax–Wendroff-type schemes in 3D. The main result of this paper is the creation of new Lax–Wendroff-type second-order accurate optimally-stable dispersive non-split scheme that is in conservation form. The scheme is created by using linear equivalence to transform a symmetrized dimensionally-split scheme (based on a one-dimensional Lax–Wendroff scheme) to conservation form. We then create both composite and hybrid schemes by combining the new scheme with the diffusive first-order accurate Lax–Friedrichs scheme. Codes based on these schemes perform well on difficult fluid flow problems.