A Kind Of Trigonometrically-fitted Runge-kutta Methods

It is well known that solutions existence of differential equation as y' = f(x, y) may be proofed under the extreme general conditions.With the time development, we don't only care about the solution existence,but also variables solution values or the approximate value corresponding to certain specific independent in some definition scopes.Such group of values are called this differential equation in this scope numerical solution, seeking the numerical solution the process called the value solution differential equation.Runge -Kutta method is very important in the ordinary differential equation value solution method.Runge-Kutta method already had 110 years history.Prom the day it was born,it had a far-reaching impact in The development of mathematics.In this 110 years of history, as it is being applied to more math problems,Runge -Kutta method itself has done a lot of improvement with the problem,many mathematicians are still in their research today.The algorithm has a history of more than one century, and has been improved and applied.It shows that this algorithm are still worth studying, improved and perfected. It will be applied to the problems which are more mysteries.This paper concisely studies the origin and history of development of Runge-Kutta methods which are very important computational methods in ordinary differential equations. It reviews some of the early contributions due to Runge, Heun and Kutta, shows that Runge-Kutta methods originated from Euler' polygon method.Finally,we describe the numerical results with the practical problems.In the article of the first chapter, we gave a briefing on the background of the issue. Because ordinary differential equations by the numerical method can be divided into two categories which are Hamilton system and non-Hamilton sys-tem,whether Hamilton system or non-Hamilton system ,Runge-Kurta methods are the most popular and most importantly.In the next chapter,we concisely studies the origin and history of development of Runge-Kutta methods, reviews some of the early contributions due to Runge, Heun and Kutta,and show that Runge-Kutta method is a special one-step method.It can be regarded as taking the number in the number of integral curve slope of the tangent point. another one (or several) Arithmetic (or weighted) average slope of the new post and then from the slope, Linear curve generation to further push forward the process.After introducing the history and origin of the Runge - Kutta method,in the next several chapters,several popular major improvements to the current Runge-Kutta method would be introduced.At first, the symplectic Runge-Kutta method is introduced(Chapter III). In recent years, many foreign scientists re-entered to the study of the Runge-Kutta method.Based on previous studies, construct that exponentially fitted Runge-Kutta method,trigonometrically fitted Runge-Kutta method, symplectic trigonometrically fitted Runge-Kutta method and so on.In the fourth chapter, we re-construct the 4-order trigonometrically-fitted Runge-Kutta method.The new 4-order trigonometrically-fitted Runge-Kutta method could be of high accuracy and convergence with a much faster rate with large step,if fitting well.At the same time,we also show that new 4-order exponentially-fitted Runge-Kutta method constructed by the same way is not convergenced to the classic 4-order Runge-Kutta method. The method can also be presented using the Butcher table below:An s-stage explicit Runge-Kutta method can be expressed by the following relations:The method is associated with the operatorwhere z is a continuously differentiable function.We want our method to integrate at each stage the the functions(sinωx) and cos(ωx) (for the trigonometrically fitted case). In the situationSo the new method has all its coefficients except for:For small values of v the coefficients are subject to heavy cancelations, thus we expand the coefficients over the Taylor series around zero. We see that for v→0 the method converges to the fouth order method.In the sixth Chapter , We would apply the all kinds of new methods to the practical problems and get the numerical results.By comparison,we can see that improved methods has great advantages in various fields.In the final chapter,by compared the numerical results,we point out the strengths and weaknesses of any methods. At last,we pointe out the possible directions for future work.