A Class of Two-Derivative Two-Step Runge-Kutta Methods for Non-Stiff ODEs

In this paper, a new class of two-derivative two-step Runge-Kutta (TDTSRK) methods for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equation (ODEs) is considered. The TDTSRK methods are a special case of multi-derivative Runge-Kutta methods proposed by Kastlunger and Wanner (1972). The methods considered herein incorporate only the first and second derivatives terms of ODEs. These methods possess large interval of stability when compared with other existing methods in the literature. The experiments have been performed on standard problems, and comparisons were made with some standard explicit Runge-Kutta methods in the literature.

[1]  R. I. Okuonghae,et al.  Second derivative general linear methods , 2013, Numerical Algorithms.

[2]  John C. Butcher,et al.  General linear methods for ordinary differential equations , 2009, Math. Comput. Simul..

[3]  Rosemary A. Renaut,et al.  Two-step Runge-Kutta methods , 1991 .

[4]  Zdzislaw Jackiewicz,et al.  A general class of two-step Runge-Kutta methods for ordinary differential equations , 1995 .

[5]  Ernst Hairer,et al.  Order Conditions for General Two-Step Runge--Kutta Methods , 1997 .

[6]  Z. Jackiewicz,et al.  Derivation and implementation of Two-Step Runge-Kutta pairs , 2002 .

[7]  T. Özis,et al.  On explicit two-derivative two-step Runge–Kutta methods , 2018, Computational and Applied Mathematics.

[8]  Raffaele D'Ambrosio,et al.  Continuous two-step Runge–Kutta methods for ordinary differential equations , 2010, Numerical Algorithms.

[9]  J. Lambert.  Numerical Methods for Ordinary Differential Equations , 1991 .

[10]  A. Abdi,et al.  Maximal order for second derivative general linear methods with Runge-Kutta stability , 2011 .

[11]  L. M. Skvortsov.  Explicit two-step Runge-Kutta methods , 2010 .

[12]  Robert P. K. Chan,et al.  On explicit two-derivative Runge-Kutta methods , 2010, Numerical Algorithms.

[13]  Z. Jackiewicz,et al.  Construction of two-step Runge--Kutta methods with large regions of absolute stability , 2003 .

[14]  Gerhard Wanner,et al.  Runge Kutta processes with multiple nodes , 1972, Computing.

[15]  John C. Butcher,et al.  Order conditions for two-step Runge-Kutta methods , 1997 .

[16]  Zdzislaw Jackiewicz,et al.  Construction of two-step Runge-Kutta methods of high order for ordinary differential equations , 2004, Numerical Algorithms.

[17]  S. Tracogna.  Implementation of two-step Runge-Kutta methods for ordinary differential equations , 1996 .