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

The refined inviscid stability condition and cellular instability of viscous shock waves

Combining work of Serre and Zumbrun, Benzoni-Gavage, Serre, and Zumbrun, and Texier and Zumbrun, we propose as a mechanism for the onset of cellular instability of viscous shock and detonation waves in a finite-cross-section duct the violation of the refined planar stability condition of Zumbrun--Serre, a viscous correction of the inviscid planar stability condition of Majda. More precisely, we show for a model problem involving flow in a rectangular duct with artificial periodic boundary conditions that transition to multidimensional instability through violation of the refined stability condition of planar viscous shock waves on the whole space generically implies for a duct of sufficiently large cross-section a cascade of Hopf bifurcations involving more and more complicated cellular instabilities. The refined condition is numerically calculable as described in Benzoni-Gavage--Serre-Zumbrun

Magnetohydrodynamic code for gravitationally-stratified media

Aims. We describe a newly-developed magnetohydrodynamic (MHD) code with the capacity to simulate the interaction of any arbitrary perturbation (i.e., not necessarily limited to the linearised limit) with a magnetohydrostatic equilibrium background. Methods. By rearranging the terms in the system of MHD equations and explicitly taking into account the magnetohydrostatic equilibrium condition, we define the equations governing the perturbations that describe the deviations from the background state of plasma for the density, internal energy and magnetic field. We found it was advantageous to use this modified form of the MHD equations for numerical simulations of physical processes taking place in a stable gravitationally-stratified plasma. The governing equations are implemented in a novel way in the code. Sub-grid diffusion and resistivity are applied to ensure numerical stability of the computed solution of the MHD equations. We apply a fourth-order central difference scheme to calculate the spatial derivatives, and implement an arbitrary Runge-Kutta scheme to advance the solution in time. Results. We have built the proposed method, suitable for strongly-stratified magnetised plasma, on the base of the well-documented Versatile Advection Code (VAC) and performed a number of one- and multi-dimensional hydrodynamic and MHD tests to demonstrate the feasibility and robustness of the code for applications to astrophysical plasmas

Pointwise Green function bounds and stability of combustion waves

Generalizing similar results for viscous shock and relaxation waves, we establish sharp pointwise Green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including both Majda's model and the full reacting compressible Navier--Stokes equations with artificial viscosity with general multi-species reaction and reaction-dependent equation of state, % under the necessary conditions of strong spectral stability, i.e., stable point spectrum of the linearized operator about the wave, transversality of the profile as a connection in the traveling-wave ODE, and hyperbolic stability of the associated Chapman--Jouguet (square-wave) approximation. Notably, our results apply to combustion waves of any type: weak or strong, detonations or deflagrations, reducing the study of stability to verification of a readily numerically checkable Evans function condition. Together with spectral results of Lyng and Zumbrun, this gives immediately stability of small-amplitude strong detonations in the small heat-release (i.e., fluid-dynamical) limit, simplifying and greatly extending previous results obtained by energy methods by Liu--Ying and Tesei--Tan for Majda's model and the reactive Navier--Stokes equations, respectively

A stability index for detonation waves in Majda's model for reacting flow

Using Evans function techniques, we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in [GZ,BSZ], yielding useful necessary conditions for stability. Here, we carry out the analysis in the context of the Majda model, a simplified model for reacting flow; the method is extended to the full Navier-Stokes equations of reacting flow in [Ly,LyZ]. The resulting stability condition is satisfied for all nondegenerate, i.e., spatially exponentially decaying, weak and strong detonations of the Majda model in agreement with numerical experiments of [CMR] and analytical results of [Sz,LY] for a related model of Majda and Rosales. We discuss also the role in the ZND limit of degenerate, subalgebraically decaying weak detonation and (for a modified, ``bump-type'' ignition function) deflagration profiles, as discussed in [GS.1-2] for the full equations.Comment: 36 pages, 3 figure