Efficient numerical stability analysis of detonation waves in ZND

As described in the classic works of Lee--Stewart and Short--Stewart, the numerical evaluation of linear stability of planar detonation waves is a computationally intensive problem of considerable interest in applications. Reexamining this problem from a modern numerical Evans function point of view, we derive a new algorithm for their stability analysis, related to a much older method of Erpenbeck, that, while equally simple and easy to implement as the standard method introduced by Lee--Stewart, appears to be potentially faster and more stable

On the shock wave spectrum for isentropic gas dynamics with capillarity

AbstractWe consider the stability problem for shock layers in Slemrod's model of an isentropic gas with capillarity. We show that these traveling waves are monotone in the weak capillarity case, and become highly oscillatory as the capillarity strength increases. Using a spectral energy estimate we prove that small-amplitude monotone shocks are spectrally stable. We also show that monotone shocks have no unstable real spectrum regardless of amplitude; this implies that any instabilities of these monotone traveling waves, if they exist, must occur through a Hopf-like bifurcation, where one or more conjugate pairs of eigenvalues cross the imaginary axis. We then conduct a systematic numerical Evans function study, which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths. In particular, we show that the transition from monotone to nonmonotone profiles does not appear to trigger any instabilities

Balanced flux formulations for multidimensional Evans function computations for viscous shocks

The Evans function is a powerful tool for the stability analysis of viscous shock profiles; zeros of this function carry stability information. In the one-dimensional case, it is typical to compute the Evans function using Goodman's integrated coordinates [G1]; this device facilitates the search for zeros of the Evans function by winding number arguments. Although integrated coordinates are not available in the multidimensional case, we show here that there is a choice of coordinates which gives similar advantages