A method for the numerical integration of ordinary differential equations

Abstract

where y(x) denotes the solution of the differential equation. The idea is to use a quadrature formula to estimate the integral of (1). This requires knowledge of the integrand at specified arguments xi in (x0, xo + h)-hence we require the values of y(x) at these arguments. A numerical integration method may be used to estimate y(x) for the required arguments. In this way a numerical integration method is combined with a quadrature formula to obtain another numerical integration method. A large number of methods may be devised, depending on which combination of quadrature formula and integration method is used. In particular, the Gauss two-point quadrature formula combined with the Runge-Kutta fourth order method appears to give excellent results [1]. We propose here the combination of the Radau three-point quadrature formula with the Runge-Kutta fourth order method. The resulting method seems to give greater accuracy with the same amount of work. 2. The Method. The Radau quadrature formula [2] gives