Simplified Second-Order Godunov-Type Methods

Harten, Lax, and van Leer showed how to construct a simple approximate Riemann solution which contains only one intermediate step. This construction assumes that you have a priori bounds on the smallest and largest signal velocities in the exact Riemann solution. Here we propose a number of algorithms for obtaining these bounds. Heuristic arguments are presented to support our choice of bounds. We derive first-order schemes which use these approximate Riemann solutions and show their relationship to known finite difference schemes. Next, we use the approach of van Leer et al. 1 to construct second-order schemes based on these approximate Riemann solutions. Of particular interest is a central difference scheme requiring no upwind switches. This scheme is only slightly more complex than standard predictor–corrector finite difference schemes. Preliminary numerical results are presented which show that these schemes are nonoscillatory, have good shock resolution and produce results which are competitive with those produced by more complex second order Godunov-type schemes.