Optimal monotonicity-preserving perturbations of a given Runge–Kutta method

Perturbed Runge–Kutta methods (also referred to as downwind Runge–Kutta methods) can guarantee monotonicity preservation under larger step sizes relative to their traditional Runge–Kutta counterparts. In this paper we study the question of how to optimally perturb a given method in order to increase the radius of absolute monotonicity (a.m.). We prove that for methods with zero radius of a.m., it is always possible to give a perturbation with positive radius. We first study methods for linear problems and then methods for nonlinear problems. In each case, we prove upper bounds on the radius of a.m., and provide algorithms to compute optimal perturbations. We also provide optimal perturbations for many known methods.Inmaculada Higueras was supported by Ministerio de Economía y Competividad, Spain, Projects MTM2014-53178-P and MTM2016-77735-C3-2-P. David I. Ketcheson and Tihamér A. Kocsis were supported by KAUST Award No. FIC/2010/05-2000000231. Tihamér A. Kocsis was also supported by TÁMOP-4.2.2.A-11/1/KONV-2012-0012: Basic research for the development of hybrid and electric vehicles, supported by the Hungarian Government and co-financed by the European Social Fund