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

摘要

Direct-correction and asymptotic successive-correction methods based on the second equivalent equation are applied to the Euler forward explicit scheme. In direct-correction, the truncation error terms of the second equivalent equation which contain higher-order derivatives together with a starting procedure, are discretized by means of finite differences. Both explicit and implicit direct-correction schemes are presented and their stability regions are studied. The asymptotic successive-correction numerical technique developed in Part II of this series with a consistent starting procedure is applied to the second equivalent equation. Both all-backward and all-centered asymptotic successive-correction methods are presented. The numerical methods introduced in this paper are applied to autonomous and non-autonomous, scalar and systems of ordinary differential equations and compared with the 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 successive-correction techniques for large time steps due to the need for higher-order derivatives of the Euler solution; however, for sufficiently small time steps, but larger enough so that round-off errors are negligible, both methods have nearly the same accuracy.