High-frequency asymptotics and 1-D stability of ZND detonations in the small-heat release and high-overdrive limits

TL;DR

This paper proves the one-dimensional spectral stability of ZND detonations in the small heat release and high overdrive limits, providing a rigorous foundation and quantitative estimates for numerical stability analysis.

Contribution

It introduces a novel rescaling and high-frequency analysis method to rigorously establish stability in these limits, confirming and extending Erpenbeck's numerical findings.

Findings

Spectral stability of ZND detonations in small heat release limit

High-frequency stability established with uniform parameter estimates

Quantitative bounds suitable for numerical stability investigations

Abstract

We establish one-dimensional spectral, or "normal modes", stability of ZND detonations in the small heat release limit and the related high overdrive limit with heat release and activation energy held fixed, verifying numerical observations of Erpenbeck in the 1960s. The key technical points are a strategic rescaling of parameters converting the infinite overdrive limit to a finite, regular perturbation problem, and a careful high-frequency analysis depending uniformly on model parameters. The latter recovers the important result of high-frequency stability established by Erpenbeck by somewhat different techniques. Notably, the techniques used here yield quantitative estimates well suited for numerical stability investigation.