Numerical Analysis Of Runge-Kutta-Nystr?m-method

Second-order ordinary differential equations are used in many fields of science and engineering,such as celestial mechanics,quantum mechanics,theoretical physics and chemistry.They usually have periodic solutions or oscillatory solutions,and this brings difficulties to obtain solutions by using numerical methods.In recent years,the study of numerical methods for second-order ordinary differential equations has received much attention.Runge-Kutta-Nystr?m(RKN)methods are among important numerical methods for solving second-order ordinary differential equations.In this thesis,RKNmethods are treated as composite linear methods,and the order conditions of the methods get from the recursive relationships.The main contents are as follows.In Chapter 1,we introduce the background and review some literature,then give our main aim of this thesis.In Chapter 2,The local discretization error of the RKN-method is defined and we express the local discrete error by using Taylor expansion.Then we establish two recursion relations for simplifying the general order conditions to orthogonal conditions.Without using the rooted tree theory and differential operator,we generate the order conditions through a simple recursive relationship.In Chapter 3,we extend the recursion in Chapter 2 with various mappings.Firstly,the definition of the recursive differential is given.We respectively give recursive expressions of recursive differential,rooted trees,elementary differential,and elementary weights by establishing various bijections.Then,the related results of the classical theory can be reproduced.In Chapter 4,an explicit RKN-method is given in detail,and its convergence result is obtained by using our theoretical results.Then,we apply this explicit RKN-method to solve a second-order ordinary differential equation.A numerical experiment is presented to confirm the theoretical results.