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The stability of travelling wave Chapman–Jouguet and moderately overdriven detonations of Zeldovich–von Neumann–Döring type is formulated for a general system that incorporates the idealized gas and condensed-phase (liquid or solid) detonation models. The general model consists of a two-component mixture with a one-step irreversible reaction between reactant and product. The reaction rate has both temperature and pressure sensitivities and has a variable reaction order. The idealized condensed-phase model assumes a pressure-sensitive reaction rate, a constant-γ caloric equation of state for an ideal fluid, with the isentropic derivative γ=3, and invokes the strong shock limit. A linear stability analysis of the steady, planar, ZND detonation wave for the general model is conducted using a normal-mode approach. An asymptotic analysis of the eigenmode structure at the end of the reaction zone is conducted, and spatial boundedness (closure) conditions formally derived, whose precise form depends on the magnitude of the detonation overdrive and reaction order. A scaling analysis of the transonic flow region for Chapman–Jouguet detonations is also studied to illustrate the validity of the linearization for Chapman–Jouguet detonations. Neutral stability boundaries are calculated for the idealized condensed-phase model for one- and two-dimensional perturbations. Comparisons of the growth rates and frequencies predicted by the normal-mode analysis for an unstable detonation are made with a numerical solution of the reactive Euler equations. The numerical calculations are conducted using a new, high-order algorithm that employs a shock-fitting strategy, an approach that has significant advantages over standard shock-capturing methods for calculating unstable detonations. For the idealized condensed-phase model, nonlinear numerical solutions are also obtained to study the long-time behaviour of one- and two-dimensional unstable Chapman–Jouguet ZND waves.

A novel, highly accurate numerical scheme based on shock-fitting coupled with high order spatial and temporal discretization is applied to a classical unsteady detonation problem to generate solutions with unprecedented accuracy. The one-dimensional reactive Euler equations for a calorically perfect mixture of ideal gases whose reaction is described by single-step irreversible Arrhenius kinetics are solved in a series of calculations in which the activation energy is varied. In contrast with nearly all known simulations of this problem, which converge at a rate no greater than first order as the spatial and temporal grid is refined, the present method is shown to converge at a rate consistent with the fifth order accuracy of the spatial discretization scheme. This high accuracy enables more precise verification of known results and prediction of heretofore unknown phenomena. To five significant figures, the scheme faithfully recovers the stability boundary, growth rates, and wave-numbers predicted by an independent linear stability theory in the stable and weakly unstable regime. As the activation energy is increased, a series of period-doubling events are predicted, and the system undergoes a transition to chaos. Consistent with general theories of non-linear dynamics, the bifurcation points are seen to converge at a rate for which the Feigenbaum constant is 4.66 ± 0.09, in close agreement with the true value of 4.669201 . . .. As activation energy is increased further, domains are identified in which the system undergoes a transition from a chaotic state back to one whose limit cycles are characterized by a small number of non-linear oscillatory modes. This result is consistent with behavior of other non-linear dynamical systems, but not typically considered in detonation dynamics.

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