Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations

Additive Runge Kutta (ARK) methods are investigated for application to the spatially discretized one dimensional convection diffusion reaction (CDR) equations. First, accuracy, stability, conservation, and dense output are considered for the general case when N different Runge Kutta methods are grouped into a single composite method. Then, implicit explicit, N = 2, additive Runge Kutta (ARK2) methods from third to fifth order are presented that allow" for integration of stiff terms by an L stable, stiffly accurate explicit, singly diagonally implicit Runge Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge Kutta method (ERK). Coupling error terms are of equal order to those of the elemental methods. Derived ARK2 methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z [z] --+ -oc, and retain high stability efficiency in the absence of stiffness, z[ I] --+ 0. Extrapolation type stage value predictors are provided based on dense output formulae. Optimized methods minimize both leading order ARK2 error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new" schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID controller. While results for the fifth order method are disappointing, both the new" third and fourth order methods are at least as efficient as existing ARK2 methods while offering error control and stage value predictors.