Studies of the accuracy of time integration methods for reaction–diffusion equations

In this study we present numerical experiments of time integration methods applied 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 mechanisms 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.