Balancing Source Terms and Flux Gradientsin High-Resolution Godunov Methods : The Quasi-Steady Wave-Propogation AlgorithmRandall

Conservation laws with source terms often have steady states in which the ux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have diiculty preserving such steady states and cannot accurately calculate small perturbations of such states. Here a variant of the wave-propagation algorithm is developed which addresses this problem by introducing a Riemann problem in the center of each grid cell whose ux diierence exactly cancels the source term. This leads to modiied Riemann problems at the cell edges in which the jump now corresponds to perturbations from the steady state. Computing waves and limiters based on the solution to these Riemann problems gives high-resolution results. The 1D and 2D shallow water equations for ow over arbitrary bottom topography are use as an example, though the ideas apply to many other systems. The method is easily implemented in the software package clawpack.