Title:        Development and Analysis of 3D Non-Split Optimally 
                Stable Lax-Wendroff Type Difference Scheme for 
                Conservation Laws
  Authors:      M. Kucharik ^1, R. Liska* ^1, S. Steinberg ^2, 
                B. Wendroff ^3

  Affiliation:  ^1 Czech Technical University, Prague, Czech Republic
                ^2 University of New Mexico, Albuquerque, U.S.A.
                ^3 Los Alamos National Laboratory, Los Alamos, U.S.A.
  Abstract:
Direct generalization of a 2D non-split optimally stable Lax-Wendroff (LW) type
finite difference scheme to 3D is unconditionally unstable.  A variation of
this scheme is sub-optimally stable.  On the other hand the 3D split extension
of the 1D optimally stable LW scheme is also optimally stable.  We start with
this optimally stable 3D split scheme, assume linearity of fluxes and derive a
new 3D non-split scheme.  As the transformation is linear and stability
analysis is done using scalar advection, the new scheme remains optimally
stable.  The new scheme is second order both for scalar advection and a general
non-linear conservation law.  Derivation and analysis of the new scheme has
involved processing of large complicated formulas on a 3D stencil (basic and
staggered) and has heavily relied on computer algebra facilities.  Computer
algebra tools have been used for scheme transformation, stability analysis,
truncation error analysis and modified equation construction.