Higher-order additive Runge–Kutta schemes for ordinary differential equations

Abstract

Two new implicit–explicit, additive Runge–Kutta (ARK2) methods are given with fourth- and fifth-order formal accuracies, respectively. Both combine explicit Runge–Kutta (ERK) methods with explicit, singly-diagonally implicit Runge–Kutta (ESDIRK) methods and include an embedded method for error control. The two methods have ESDIRKs which are internally L-stable on stages three and higher, have only modestly negative eigenvalues to the stage and step algebraic-stability matrices and have stage-order two. To improve computational efficiency, the fourth-order method has a diagonal coefficient of 0.1235. This is done to offset much of the extra computational cost of an extra stage by facilitating iterative convergence at each stage. Linear stability domains for both ERK methods have been made quite large and the dominant coupling stability term between the stability of the ESDIRK and ERK for very stiff modes has been removed. Though the fourth-order method is one of the best all-around fourth-order IMEX ARK2 of which we are aware, the fifth-order method is likely best suited to mildly stiff problems with tight error tolerances. Methods are tested using the Van der Pol and Kaps' singular-perturbation problems. Results suggest that these new methods represent an improvement over existing methods of the same class.