Comparison of some recent numerical methods for initial-value problems for stiff ordinary differential equations

We consider the combustion equation as one of the candidates from the class of stiff ordinary differential equations. A solution over a length of time that is inversely proportional to δ > 0 (where δ > 0 is a small disturbance of the pre-ignition state) is sought. This problem has a transient at the midpoint of the integration interval. The solution changes from being non-stiff to stiff, and afterwards becomes non-stiff again. We provide its asymptotic and numerical solution obtained via a variety of methods. Comparisons are made for the numerical results which we obtain with the MATLAB ode solvers (ode45, ode15s and ode23s) and some nonstandard finite difference methods. Results corresponding to standard finite difference method are also presented. Furthermore, the discussion on these approaches along with the others, provides several open problems for new and young researchers. c © 2007 Elsevier Ltd. All rights reserved.