Asymptotic and Numerical Analysis of Dynamics in a Generalized Free-Interfacial Combustion Model

Abstract

Dynamics of temperature distribution and interfacial front propagation in a generalized solid combustion model are studied through both asymptotic and numerical analyses. For asymptotic analysis, we focus on the weakly nonlinear case where a small perturbation of a neutrally stable parameter is taken so that the linearized problem is marginally unstable. Multiple scale expansion method is used to obtain an asymptotic solution for large time by modulating the most linearly unstable mode. On the other hand, we integrate numerically the exact problem by the Crank-Nicolson method. Since the numerical solutions are very sensitive to the derivative interfacial jump condition, we integrate the partial differential equation to obtain an integral-differential equation as an alternative condition. The result system of nonlinear algebraic equations is then solved by the Newton’s method, taking advantage of the sparse structure of the Jacobian matrix. Finally, we show that our asymptotic solution captures the marginally unstable behaviors of the solution for a range of model parameters.