Conservative, shock-capturing transport methods with nonconservative velocity approximations

Conservative high-resolution, or shock-capturing, methods have become widel y used for modeling transport equations described by conservation laws. In many geoscience applications, the transport equation is coupled to a conser vation or continuity equation for a velocity field. Depending on how the ve locity 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 accoun ts for the fact that the velocity may be nonconservative. We present severa l numerical examples and prove stability bounds and an a priori error estim ate for the corrected method.