Efficient construction of high-resolution TVD conservative schemes for equations with source terms:: application to shallow water flows

High-resolution total variation diminishing (TVD) schemes are widely used for the numerical approximation of hyperbolic conservation laws. Their extension to equations with source terms involving spatial derivatives is not obvious. In this work, efficient ways of constructing conservative schemes from the conservative, non-conservative or characteristic form of the equations are described in detail. An upwind, as opposed to a pointwise, treatment of the source terms is adopted here, and a new technique is proposed in which source terms are included in the flux limiter functions to get a complete second-order compact scheme. A new correction to fix the entropy problem is also presented and a robust treatment of the boundary conditions according to the discretization used is stated. Copyright (C) 2001 John Wiley & Sons, Ltd.