On the method of modified equations. IV. Numerical techniques based on themodified equation for the Euler forward difference method

The modified equation method is studied as a technique for the development of new numerical techniques for ordinary differential schemes based on the third modified or (simply) modified equation of the explicit Euler forward method. Both direct-correction and successive-correction techniques based o n the modified equation are used to obtain higher-order schemes. The result ing numerical techniques are completely explicit, of order of accuracy as h igh as desired, and self-starting since the truncation error terms in the m odified equation have no derivatives. The methods introduced in this paper are applied to autonomous and non-autonomous, scalar and systems of ordinar y differential equations and compared with second- and fourth-order accurat e Runge-Kutta schemes. It is shown that, for sufficiently small step sizes, the fourth-order direct-correction and successive-correction methods are a s accurate as the fourth-order Runge-Kutta scheme. (C) 1999 Elsevier Scienc e Inc. All rights reserved.