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

Direct-correction and asymptotic successive-correction methods based on the second equivalent equation are applied to the Euler forward explicit schem e. In direct-correction, the truncation error terms of the second equivalen t equation which contain higher-order derivatives together with a starting procedure, are discretized by means of finite differences. Both explicit an d implicit direct-correction schemes are presented and their stability regi ons are studied. The asymptotic successive-correction numerical technique d eveloped in Part II of this series with a consistent starting procedure is applied to the second equivalent equation. Both all-backward and all-center ed asymptotic successive-correction methods are presented. The numerical me thods introduced in this paper are applied to autonomous and non-autonomous , scalar and systems of ordinary differential equations and compared with t he results of second- and fourth-order accurate Runge-Kutta methods. It is shown that the fourth-order Runge-Kutta method is more accurate than the su ccessive-correction techniques for large time steps due to the need for hig her-order derivatives of the Euler solution; however, for sufficiently smal l time steps, but larger enough so that round-off errors are negligible, bo th methods have nearly the same accuracy. (C) 1999 Elsevier Science Inc. Al l rights reserved.