Asymptotic Solution of Stiff PDEs with the CSP Method: The Reaction Diffusion Equation

Abstract

The computational singular perturbation (CSP) method is employed for the solution of stiff PDEs and for the acquisition of the most important physical understanding. The usefulness of the method is demonstrated by analyzing a transient reaction-diffusion problem. It is shown that in the regions where the solution exhibits smooth spatial slopes, a simple nonstiff system of equations can be used instead of the full governing equations. From the simplified system, which is numerically provided by CSP and whose structure varies with space and time, important physical information comes to light. The relation of this method to the class of asymptotic expansion methods is explored. It is shown that the CSP results are identical to the ones obtained by the asymptotic methods. The identifications of the nondimensional parameters and the tedious manipulations needed by the asymptotic methods are performed by programmable numerical or analytic computations specified by CSP. Preliminary numerical results are presented validating the theoretical aspects of the proposed algorithm and providing a measure of its usefulness and its accuracy.