Computations of Slowly Moving Shocks

Abstract

Computations of slowly moving shocks by shock capturing schemes may generate oscillations that appear as a wavy tail attached to the shock front. These oscillations are generated already by first-order schemes, but become more pronounced in higher-order schemes due to their lower dissipation. We focus on two first-order schemes which seem to exhibit different behaviors: (i) the first-order upwind (UW) scheme which generates strong oscillations and (ii) the Lax–Friedrichs scheme which appears not to generate any disturbances at all. A key observation is that in the UW case, the numerical viscosity in the shock family vanishes inside the slow shock layer. Simple scaling arguments show that third-order effects on the solution may no longer be neglected. We derive the third-order modified equation for the UW scheme and regard the oscillatory solution as a traveling wave solution of the parabolic modified equation plus a small perturbation. We then look at the governing equation for the perturbation, which points to a plausible mechanism by which postshock oscillations are generated. It contains a third-order source term that becomes significant inside the shock layer, and a nonlinear coupling term which projects the perturbation on all characteristic fields, including those not associated with the shock family.