Numerical simulations of pulsating detonations: I. Nonlinear stability of steady detonations

Very-long-time numerical simulations of an idealized pulsating detonation w ith one irreversible reaction having an Arrhenius form are performed using a hierarchical adaptive second-order Godunov-type scheme. The initial data are given by the steady solution and the truncation error produces the pert urbation to trigger the instability. The detonation is allowed to run for t housands of half-reaction times of the underlying steady wave to ensure tha t the final amplitudes and periods of the nonlinear oscillations are achiev ed Thorough resolution studies are performed for various representative reg imes of the instability. It is shown that to obtain quantitatively goad sol utions over 50 numerical grid points in the half-reaction length of the ste ady detonation are required, while to obtain a converged solution over 100 points are required, even near the stability boundary. This is much higher resolution than has generally been used in previous papers in either one or two dimensions. Resolutions of less than approximately 20 points per half- reaction length give very poor Predictions of the periods and amplitudes ne ar the stability boundary or entirely spurious solutions for more unstable detonations. The evolution of the converged solutions as the activation ene rgy increases, and the detonation becomes more unstable, is also investigat ed.