A numerical stability investigation of strong ZND detonations for Majda’s model

We carry out a systematic numerical stability analysis of ZND detonations of Majda’s model with Arrhenius-type ignition function, a simplified model for reacting flow, as heat release and activation energy are varied. Our purpose is, first, to answer a question of Majda whether oscillatory instabilities can occur for high activation energies as in the full reacting Euler equations, and, second, to test the efficiency of various versions of a numerical eigenvalue-finding scheme suggested by Humpherys and Zumbrun against the standard method of Lee and Stewart. Our results suggest that instabilities do not occur for Majda’s model with Arrhenius-type ignition function, nor with a modified Arrhenius-type ignition function suggested by Lyng–Zumbrun, even in the high-activation energy limit. We find that the algorithm of Humpherys–Zumbrun is in the context of Majda’s model 100-1, 000 times faster than the one described in the classical work of Lee and Stewart and 1-10 times faster than an optimized version of the Lee–Stewart algorithm using an adaptive-mesh ODE solver.