Analysis and correction of errors generated by slowly moving shocks

In problems where shock waves move slowly relative to the grid, numerical errors in density and momentum may occur at the shock and propagate downstream, thereby contaminating the solution. These errors are analyzed in the present work and a fix is proposed. The errors are divided into two classes: start-up errors and post-shock oscillations. It is shown that a traveling wave analysis of the isothermal Euler equations describes the spike formation in the momentum for the first-order accurate Lax-Friedrichs solver very well. By smearing the density and momentum profiles appropriately, the corresponding downstream-propagating wave can be removed. The post-shock oscillations have been characterized; as a fix for this problem, a lower bound on the wave speed of the first-order accurate HLL solver is set in the artificial viscosity coefficient for density in the continuity equation. The implementation to higher-order accurate methods is not straightforward; an additional fix is proposed, which is performs well on the full multi-dimensional Euler equations.