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

  • Authors:

  • Milan Kuchařík;Richard Liska;Stanly Steinberg;Burton Wendroff

  • Affiliations:

  • Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic;Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic;Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM;Group T-7, Los Alamos National Laboratory, Los Alamos, NM

  • Venue:
  • Applied Numerical Mathematics
  • Year:
  • 2006

Quantified Score

Hi-index 0.00

Visualization

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-Wendoff-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.