Singly-Implicit Runge--Kutta Methods for Stiff, Ordinary Differential-Equations

Singly-Implicit Runge–Kutta (SIRK) methods are designed for integrating systems of stiff, first-order
ordinary differential equations (ODEs). The accuracy and stability properties of the new SIRKs are similar to those of the Radau IIA methods, at potentially reduced implementation costs. Nine stiffly-accurate, L-stable and internally L-stable SIRK methods, having stage-orders of three or four, with orders three through six are given, each with an embedded method. A stage-order five method is included as well. The accuracies of the new SIRKs and Radau IIA methods, are compared on three singular perturbation problems. Order reduction of the SIRKs is generally minimal and they perform nearly as well as Radau IIA methods. There is evidence that SIRKs can be implemented at costs commensurate with diagonally implicit methods (i.e, DIRK/SDIRK/ESDIRK), making them highly competitive integrators. Efficient implementation of SIRKs at scale however, is not reported herein.