LOOK-AHEAD LINEAR MULTISTEP METHODS FOR INITIAL VALUE PROBLEM IN ODINARY DIFFERENTIAL EQUATIONS

LOOK-AHEAD LINEAR MULTISTEP METHODS

Abstract

We are concerned with the numerical solutions of initial-value problems of ordinary differential equations. There are many existing numerical  methods for the solution of such problems, among which the discrete variable methods, like the Euler, the Runge-Kutta and the linear multistep methods are most popular because of their flexible capability for the solution of such problems. On the other hand, numerical analysts are still pursuing new  methods with better performance and higher reliability. From this point of view, we try to find more for the numerical solution of ordinary differential equations.
        In the present paper, we propose a new  class of linear multistep methods, called ``look-ahead'' linear multistep methods, which has a potential of wide applications. Here we survey the preceding works to study the methods both theoretically and practically.