Three-step Runge-Kutta Methods In Ordinary Differential Equations

Ordinary differential equations (ODEs) are widely used in the fields of physics, biology, medicine and many other engineering fields, its mathematical formulation attribute to ordinary differential equations for solutions. Many partial differential equations can be translated into the approximate solution of ordinary differential equations through discreting the quantity. There, studying the numerical solution of ordinary differential equations is very important. Now the issue of ordinary differential equations for the numerical solution is more and more mature and complete. The workers of numerical analysis constructed many practical value methods. This paper is to construct and research a new numerical method.At the beginning of this paper, we explain the background of the Runge-Kutta method(including single-step and two-step Runge-Kutta method), review the Order conditions and stability theory of Runge-Kutta method in the initial value problems.In the second chapter, we first introduce the order conditions of Runge-Kutta methods, then construct a new type of three-step Runge-Kutta methods and study its zero-stable. We main derive the order conditions of three-step Runge-Kutta method using the A-method of Albrecht.In the third chapter, we aim at that type of three-step Runge-Kutta methods, make use of the order conditions deduced in chapter II , construct three-step Runge-Kutta method of 3-order, searching the method with the calculator to have large regions of absolute stability, draw the region of stability and carry on a comparison with three-step Adams methods.In the last chapter, we carry on the number experiment to the new method, and verify the anterior result.