Some Kinds Of Exponentially Fitted Rosenbrock Methods For Solving Second-Order Ordinary Differential Equations

The initial value problems of second-order ordinary differential equations have attract-ed much attention of scholars.Second-order ordinary differential equations often appear in the fields of celestial mechanics,theoretical physics and etc.Their solutions are often exponential or oscillatory.Therefore,the research on numerical methods of second-order ordinary differential equations has been paid attention by many scholars at home and abroad.Many effective numerical methods have been proposed in the study of second-order ordi-nary differential equations.There are a lot of research results of Runge-Kutta-Nystrom methods.In addition,Rosenbrock methods and Rosenbrock-Nystrom methods,which be-long to semi-implicit methods,have attracted wide attention of scholars because of their less computational complexity,easy implementation and better stability.For a class of ini-tial value problems of ordinary differential equations whose solutions can be expressed as a linear combination of {e?1x,e?2x,···}(?i?C,i=1,2,…),exponential fitting methods are often very effective.The idea of exponential fitting can be applied to the existing tra-ditional methods to obtain a new methods which is more advantageous than the traditional methods.Common exponential methods for second-order ordinary differential equation-s include exponential fitting Runge-Kutta-Nystrom methods,exponential fitting two-step hybrid methods,exponential fitting block methods,etc.But the exponential fitting Rosen-brock methods are rarely studied.In this paper,the idea of exponential fitting is extended to Rosenbrock-Nystrom methods.At the same time,the Rosenbrock methods for solving first-order ordinary differential equations are applied to second-order ordinary differential equations.By making full use of the characteristics of the system itself,several exponen-tial fitting Rosenbrock methods for solving second-order ordinary differential equations are constructed,so that the constructed methods can better approch to the solution of the system.In Chapter 1,the background of initial value problems of second-order ordinary differ-ential equations and the numerical methods are introduced,the development and improve-ment of the Rosenbrock methods by the scholars are reviewed.In Chapter 2,the Rosenbrock-Nystrom methods for solving second-order ordinary d-ifferential equations are combined with the idea of exponential fitting.The corresponding exponential fitting Rosenbrock-Nystrom methods are constructed,then,the dispersion error and dissipation error of this kind of exponential fitting methods are analyzed.In Chapter 3,the second-order ordinary differential equations are transformed into the first-order ordinary differential equations,the traditional exponential fitting Rosenbrock methods are applied to the second-order ordinary differential equation.Two kinds of expo-nential fitting Rosenbrock methods for solving the second-order ordinary differential equa-tions are given,the corresponding dispersion analysis and dissipation analysis are given.In Chapter 4,numerical experiments are carried out to verify the effectiveness of the ex-ponential fitting Rosenbrock-type methods of second-order ordinary differential equations.Several groups of numerical experiments are used to compare the precision and calculation time of the methods.The experimental results are in good agreement with the theoretical results.In the last chapter,the main purpose is to make a summary of our work,and give some ideas for the incomplete research,methods and models that can be further explored.