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Milan Kucharik, Richard Liska, Stanly Steinberg and Burton Wendroff, "Optimally-stable second-order accurate difference schemes for nonlinear conservation laws in 3D", Applied Numerical Mathematics, vol. 56, pp. 589--607, 2006


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.

BibTeX Entry

author = {Milan Kucharik and Richard Liska and Stanly Steinberg and Burton Wendroff},
title = {Optimally-stable second-order accurate difference schemes for nonlinear conservation laws in 3D},
year = {2006},
urlpdf = {http://math.lanl.gov/~kucharik/Papers/Kucharik-Liska-Steinberg-Wendroff-06.pdf},
journal = {Applied Numerical Mathematics},
volume = {56},
pages = {589--607}