Second Derivative Mono-Implicit Runge-Kutta Methods

Abstract

Mono-implicit Runge-Kutta (MIRK) methods are Runge-Kutta methods having its stages depending on its output. In this paper, we develop a family of second derivative mono-implicit Runge-Kutta (SDMIRK) methods for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The SDMIRK methods are extension of the MIRK having first and second derivative terms. The general order conditions for the stages and output methods are presented. The SDMIRK methods for stages s = 3 and s = 4 derived were found to be A-stable, while methods for s = 5 and s = 6 are A(α)-stable. Implementation procedures and numerical experiment are discussed herein. Results obtained by the SDMIRK method are favourable than the results of second derivative backward difference formular (SDBDF) and second derivative linear multistep method (SDLMM).