Diagonally-Implicit Runge-Kutta-Nystr(?)m Methods For The Second-order Ordinary Differential Equations

The initial value problems of second-order ordinary differential equations often arise in many scientific and engineering fields,such as celestial mechanics,quantum mechanics,theoretical physics and chemistry,electronics and the semi-discretization of the partial differential equations,etc.Generally,the initial value problems of second-order ordinary differential equations have periodic solutions or oscillate solutions, it brings difficulties for numerical solving.So,recently,the numerical methods for the second-order ordinary differential equations were of concern and we have got a large number of research results.Runge-Kutta-Nystr(o|¨)m methods are commonly used to sovle the second-order ordinary differential equations.This paper mainly consider diagonally-implicit Runge-Kutta-Nystr(o|¨)m methods,this kind of methods has good stability,lower calculate amounts and can achieve high order when applied to the second-order ordinary differential equations.This paper contains two parts.In the first part,we study the R-stability and P-stability of two-stage diagonally-implicit Runge-Kutta-Nystr(o|¨)m methods of order three for the second-order ordinary differential equations,and obtain the corresponding R-stable interval and we show that the P-stable two-stage Runge-Kutta-Nystr(o|¨)m methods of order three do not exist.In the second part,we study the dispersive order of two-stage diagonally-implicit Runge-Kutta-Nystr(o|¨)m methods of order three for the second-order ordinary differential equations,and construct some methods of phase-lag orders 6 and 8.These extend the discussion of R-stability,P-stability and dispersive order of singly diagonally-implicit Runge-Kutta-Nystr(o|¨)m methods to diagonally-implicit Runge-Kutta-Nystr(o|¨)m methods.