كنترل نقص روش رونگ - كوتا با استفاده از درونيابي

Defect Control for RK-methods using Computed Interpolation

The quest for reliable integration of initial value problems (IVPs) for ordinary differential equations (ODEs) is an important and popular topic in numerical analysis. We limit ourselves to the first order initial value problems because we can change each mth order differential equation to a system of first order differential equations. The class of methods we address in this paper is the class of continuous Runge-Kutta methods (CRKs), that combines discrete Runge-Kutta methods with Hermite-Birkhoff interpolation and gives us the continuous approximate solution. In this situation, by using the defect error to control error and some changes in the base of interpolations, we can derive a method that is almost independent of the problem integrated. This makes it possible for us to have an accurate and local estimation of the error independent of the equation, when we can further use it in determining variable step-size in adaptive methods. The error analysis in numerical results clarifies the details of this issue.