Computing with high-resolution upwind schemes for hyperbolic equations

Computational aspects of modern high-resolution upwind finite-difference schemes for hyperbolic systems of conservation laws are examined. An operational unification is demonstrated for constructing a wide class of flux-difference-split and flux-split schemes based on the design principles underlying total variation diminishing (TVD) schemes. Consideration is also given to TVD scheme design by preprocessing, the extension of preprocessing and postprocessing approaches to general control volumes, the removal of expansion shocks and 'glitches', relaxation methods for implicit TVD schemes, and a new family of high-accuracy TVD schemes.