Conservative, shock-capturing transport methods with nonconservative velocity approximations

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.