Diagonally implicit Runge–Kutta methods for stiff ODEs

Abstract

Based principally on a recent review of diagonally implicit Runge–Kutta (DIRK) methods applied to stiff first-order ordinary differential equations (ODEs) by the present authors, several nearly optimal, general purpose, DIRK-type methods are presented. Methods given range from third- to sixth-order in four- to nine-stages. All are both stiffly-accurate and L-stable and many are internally L-stable on stages where this is possible. Focus is placed on stage-order two methods. To facilitate step-size control via error estimation, an embedded method is included with each method listed. As the present methods must make no concessions to an explicit method as part of an additive Runge–Kutta method, the present ESDIRKs are generally better than their IMEX analogs. Five new explicit first-stage, singly diagonally-implicit Runge–Kutta (ESDIRK) methods are presented based on lessons learned from the review. To the best of our knowledge, the sixth-order, L-stable, stage-order two, 6(5)-pair, ESDIRK included herein is the only one of its kind.