Numerical methods for a coupled system of differential equations arising from a thermal ignition problem

This article is concerned with monotone iterative methods for numerical solutions of a coupled system of a first‐order partial differential equation and an ordinary differential equation which arises from fast‐igniting catalytic converters in automobile engineering. The monotone iterative scheme yields a straightforward marching process for the corresponding discrete system by the finite‐difference method, and it gives not only a computational algorithm for numerical solutions of the problem but also the existence and uniqueness of a finite‐difference solution. Particular attention is given to the “finite‐time” blow‐up property of the solution. In terms of minimal sequence of the monotone iterations, some necessary and sufficient conditions for the blow‐up solution are obtained. Also given is the convergence of the finite‐difference solution to the continuous solution as the mesh size tends to zero. Numerical results of the problem, including a case where the continuous solution is explicitly known, are presented and are compared with the known solution. Special attention is devoted to the computation of the blow‐up time and the critical value of a physical parameter which determines the global existence and the blow‐up property of the solution. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013