Pseudo-Runge-Kutta Methods Involving Two Points

Abstract

A third order two step method and a fourth order two step method for the numerical solution of the vector initial value problem dy ÷ dx =F(y), y( a ) = n can be defined by making evaluations of F similar to those found in a classical Runge-Kutta formula. These two step methods are different from classical Runge-Kutta methods in that evaluations of F made at the previous point are used along with those made at the current point in order to obtain the solution at the next point. If the stepsize is fixed, this use of previous computations makes it possible to obtain the solution at the next point by evaluating F two or three times for the third or fourth order method, respectively. These methods are consistent with the initial value problem and are shown to be convergent with its unique solution under certain restrictions. The local truncation error terms are given. Finally, a few numerical results are presented.