Higher-order additive Runge-Kutta schemes for ordinary differential equations (Q1633325)

The paper considers the numerical approximation of a system of ODEs where the right-hand side can be expressed as a sum of \(N\geq 2\) terms. The numerical scheme assumes a different integrator for each term. In particular, the authors develop two new additive Runge-Kutta (ARK\(_2\)) methods of order four and five, in an IMEX (implicit-explicit) formulation, for the case \(N=2\). The proposed numerical approach combines explicit Runge-Kutta (ERK) methods with explicit, singly-diagonally implicit (ESDIRK) methods, with error and step-size control implemented through a PID controller. The main features of the methods are discussed, especially those related to order conditions, error, linear stability and step-size control. The new methods are tested using two singular perturbation problems: Van der Pol equation and Kaps problem. The numerical tests indicate that the proposed methods present improvements in comparison to other existing methods of the same class.