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

摘要

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 on the modified equation are used to obtain higher-order schemes. The resulting numerical techniques are completely explicit, of order of accuracy as high as desired, and self-starting since the truncation error terms in the modified equation have no derivatives. The methods introduced in this paper are applied to autonomous and non-autonomous, scalar and systems of ordinary differential equations and compared with second- and fourth-order accurate Runge–Kutta schemes. It is shown that, for sufficiently small step sizes, the fourth-order direct-correction and successive-correction methods are as accurate as the fourth-order Runge–Kutta scheme.