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Conservative, shock-capturing transport methods with nonconservative velocity approximations. (English) Zbl 0964.76061
Summary: Conservative high-resolution, or shock-capturing, methods have become widely used for modeling transport equations described by conservation laws. In many geoscience applications, the transport equation is coupled to a conservation or continuity equation for a velocity field. Depending on how the velocity is approximated, the continuity equation may or may not be satisfied, either locally or globally. In this paper, we discuss the effect this has on a typical high resolution scheme, and propose a correction which accounts for the fact that the velocity may be nonconservative. We present several numerical examples and prove stability bounds and an a priori error estimate for the corrected method.

76M20 Finite difference methods applied to problems in fluid mechanics
76N15 Gas dynamics, general
76L05 Shock waves and blast waves in fluid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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