On the generation of mono-implicit Runge-Kutta-Nyström methods by mono-implicit Runge-Kutta methods

Abstract

Mono-implicit Runge-Kutta methods can be used to generate implicit Runge-Kutta-Nystr6m (IRKN) methods for the numerical solution of systems of second-order differential equations. The paper is concerned with the investigation of the conditions to be fulfilled by the mono-implicit Runge-Kutta (MIRK) method in order to generate a mono-implicit Runge-Kutta-Nystr6m method (MIRKN) that is P-stable. One of the main theoretical results is the property that MIRK methods (in standard form) cannot generate MIRKN methods (in standard form) of order greater than 4. Many examples of MIRKN methods generated by MIRK methods are presented.

Second-Derivative Two-Step Mono-Implicit Runge-Kutta Method for Stiff ODEs

“…For example, fixing m = 1, and v1 = 0 in (3) gives W1 = wn (15) Similarly, we obtain the output method of order p = 1 in (4) for s = 1 . That is wn+1 = wn + hf(xn,W1) (16) The tableau for ( 16) is (17) The method in ( 15) and ( 16) is an explicit Euler's method, which is not of interest in this paper but such schemes are suitable for non-stiff ODEs. The Euler's scheme has an interval of absolute stability of [-2,0].…”

“…Cash and Singhal [14] introduced a promising class of Mono Implicit Runge-Kutta (MIRK) method which is a subclass of method in [8] for the numerical solution of ordinary differential equations. These methods in [14] were further investigated by Muir and Owren [24], Burrage et al [7], De Meyer et al [17], Muir and Adams [23] and Dow [18] among others. A major advantage of MIRK methods over other IRK method is that it is very cheap to implement in term of the number of non-linear equations to be solved.…”

A-stable Two Derivative Mono-Implicit Runge-Kutta Methods for ODEs

“…Recently, [5] presented an Extended Mono-Implicit Runge-kutta (EMIRK) method for solving ordinary differential equations (ODEs) which is build upon the existing Mono-implicit Runge-Kutta (MIRK) methods in [ [8], [26], [9], [19], [25], [20]]. The general format of the method discussed in [5] is…”