Studies of the Accuracy of Time Integration Methods for Reaction-Diffusion Equations ∗ David L. Ropp † , John N. Shadid ‡ , and Curtis C. Ober § 20th May 2005 Abstract In this study we present numerical experiments of time integration methods ap- plied to systems of reaction-diffusion equations. Our main interest is in evaluating the relative accuracy and asymptotic order of accuracy of the methods on problems which exhibit an approximate balance between the competing component time scales. Nearly balanced systems can produce a significant coupling of the physical mecha- nisms and introduce a slow dynamical time scale of interest. These problems provide a challenging test for this evaluation and tend to reveal subtle differences between the various methods. The methods we consider include first- and second-order semi- implicit, fully-implicit, and operator-splitting techniques. The test problems include a prototype propagating nonlinear reaction-diffusion wave, a non-equilibrium radiation- diffusion system, a Brusselator chemical dynamics system and a blow-up example. In this evaluation we demonstrate a “split personality” for the operator-splitting methods that we consider. While operator-splitting methods often obtain very good accuracy, they can also manifest a serious degradation in accuracy due to stability problems. 1 Introduction In this paper we consider the numerical solution of time dependent, nonlinear, partial dif- ferential equations (PDEs) of reaction-diffusion problems. These equations are a subset of more general systems of coupled, highly nonlinear PDEs that exhibit solutions with multiple time and length scales [2][3][11]. Specifically we are interested in reaction-diffusion systems * This work was partially supported by ASCI program and the DOE Office of Science MICS program at Sandia National Laboratory. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. † Computational Mathematics & Algorithms Department, MS 1110, P.O. Box 5800, Sandia National Laboratories, Albuquerque NM, 87185-1110 (dlropp@sandia.gov) ‡ Computational Science Department, MS 1111, P.O. Box 5800, Sandia National Laboratories, Albu- querque NM, 87185-1111 (jnshadi@sandia.gov) § Computational Science Department, MS 0316, P.O. Box 5800, Sandia National Laboratories, Albu- querque NM, 87185-0316 (ccober@sandia.gov) 1