Numerical solutions of Gelfand equation in steady combustion process

Abstract

In this paper, we study a numerical algorithm to find all solutions of Gelfand equation. By utilizing finite difference discretization, the model problem defined on bounded domain with Dirichlet condition is converted to a nonlinear algebraic system, which is solved by cascadic multigrid method combining with Newton iteration method. The key of our numerical method contains two parts: a good initial guess which is constructed via collocation technique, and the Newton iteration step is implemented in cascadic multigrid method. Numerical simulations for both one-dimensional and two-dimensional Gelfand equations are carried out which demonstrate the effectiveness of the proposed algorithm. We find that by using the symmetry property of equation, numerical solutions can be obtained by mirror reflection after solving model problem in a sub-domain. This will save considerable time consumption and storage cost in computational process of cascadic multigrid method.