Computing the refined stability condition

TL;DR

This paper introduces two new methods for computing the effective viscosity coefficient that determines shock wave stability transitions, refining classical inviscid analysis by incorporating viscous effects.

Contribution

It proposes and implements two novel approaches for calculating the viscosity coefficient, enhancing the understanding of stability loss in shock waves.

Findings

Developed two practical computational methods for the viscosity coefficient.

Derived an exact solution in a specific case.

Clarified the role of viscous effects in stability transitions.

Abstract

The classical (inviscid) stability analysis of shock waves is based on the Lopatinski determinant, \Delta---a function of frequencies whose zeros determine the stability of the underlying shock. A careful analysis of \Delta\ shows that in some cases the stable and unstable regions of parameter space are separated by an open set of parameters. Zumbrun and Serre [Indiana Univ. Math. J., 48 (1999) 937--992] have shown that, by taking account of viscous effects not present in the definition of \Delta, it is possible to determine the precise location in the open, neutral set of parameter space at which stability is lost. In particular, they show that the transition to instability under suitably localized perturbations is determined by an "effective viscosity" coefficient. Here, in the simplest possible setting, we propose and implement two new approaches toward the practical computation of this coefficient. Moreover, in a special case, we derive an exact solution of the relevant differential equations.