Euler vs. Lagrange: The role of coordinates in practical Evans-function computations

TL;DR

This paper examines how the choice of coordinate systems affects Evans-function computations for viscous shock stability, introduces pseudo-Lagrangian coordinates to address computational challenges, and demonstrates their effectiveness in 2D gas dynamics.

Contribution

It reveals the impact of coordinate choice on Evans-function calculations and proposes pseudo-Lagrangian coordinates as a novel solution for multidimensional stability analysis.

Findings

Coordinate choice influences Evans-function zero detection.

Pseudo-Lagrangian coordinates improve computational stability.

Effective in 2D isentropic gas dynamics.

Abstract

The Evans function has become a standard tool in the mathematical study of nonlinear wave stability. In particular, computation of its zero set gives a convenient numerical method for determining the point spectrum of the associated linear operator (and thus the spectral stability of the wave in question). We report on an unexpected complication that frustrates this computation for viscous shock profiles in gas dynamics. Although this phenomenon---related to the choice of Eulerian or Lagrangian coordinate system used to describe the gas---is present already in the one-dimensional setting, its implications are especially important in the multidimensional case where no computationally viable Lagrangian description of the gas is readily available. We introduce new "pseudo-Lagrangian" coordinates that allow us to overcome this difficulty, and we illustrate the utility of these coordinates in the setting of isentropic gas dynamics in two space dimensions.