Additive Runge–Kutta schemes for convection–diffusion–reaction equations

Abstract

Additive Runge–Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection–diffusion–reaction (CDR) equations. Accuracy, stability, conservation, and dense-output are first 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 (ARK 2) 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 of the partitioned method are of equal order to those of the elemental methods. Derived ARK 2 methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z [ I] →−∞, 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 ARK 2 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 ARK 2 methods.