On postshock oscillations due to shock capturing schemes in unsteady flows

In this paper, the issue of postshock oscillations generated by shock-capturing schemes is investigated. Although these oscillations are frequently small enough to be ignored, there are contexts such as shock-noise interaction where they might prove very intrusive. Numerical experiments on simple nonlinear 2 x 2 systems of conservation laws are found to refute some earlier conjectures on their behavior. The trajectories in phase space of a computed state passing through a captured shock suggest the underlying mechanism that creates these oscillations. The results reveal a flaw in the way that the concept of monotonicity is extended from scalar conservation laws to systems; schemes satisfying this formal condition fail to prevent oscillations from being generated, even for monotone initial data. This indicates that satisfactory design criteria do not exist at the present time that would ensure captured shocks that are both narrow and free from oscillations. (C) 1997 Academic Press