Stability And Phase-lag Properties Of Mono-implicit Runge-kutta-nystro¨m Methods For Second-order Stiff Ordinary Differential Equations

Second-order stiff ordinarry differential equations arise in many scientific fields,the numerical solution for these equtions is difficult because of the stiffness and oscillation,which attract attention in the world.And many research results have been obtained in this field.Among all of the methods for second-order ordinary differential equations, Runge-Kutta-Nystrom (RKN)methods are commonly used.This paper primarily studies mono-implicit Runge-Kutta-Nystrom methods.The computation cost of this kind of methods has more advantages than general implicit RKN methods.This paper points at two-stage mono-implicit Runge-Kutta-Nystrom(MIRKN) methods for the second-order stiff ordinary differential equations which are mainly done in the following areas of work:R-stability,P-stability and phase-lag.Until now, we did't find any results about R-stability of mono-implicit RKN methods in the world.In this paper,we obtain the following four results.(1)We construct the R-stable two-stage MIRKN methods of order three when v=c;(2)We prove that the two-stage three-order P-stable MIRKN methods of the first class don't exist when v=c;(3)We construct P-stable two-stage second-order MIRKN methods of the second class when u=c and w≠c;(4)We consider the phase-lag of the above two-stage MIRKN methods.