Numerical testing of the stability of viscous shock waves

Abstract

A new theoretical Evans function condition is used as the basis of a numerical test of viscous shock wave stability. Accuracy of the method is demonstrated through comparison against exact solutions, a convergence study, and evaluation of approximate error equations. Robustness is demonstrated by applying the method to waves for which no current analytic results apply (highly nonlinear waves from the cubic model and strong shocks from gas dynamics). An interesting aspect of the analysis is the need to incorporate features from the analytic Evans function theory for purposes of numerical stability. For example, we find it necessary, for numerical accuracy, to solve ODEs on the space of wedge products.