On the method of modified equations. III: Numerical techniques based on the second equivalent equation for the Euler forward difference method (Q1294223)

The paper is the third of a series dealing with the assessment of modified equation methods as a means for the analysis of existing finite difference schemes and for the development of new numerical ones. [For parts II and IV see ibid. 103, No.~2-3, 141-177 and 213-240 (1999; reviewed above and below).] The validity of the modified equation based on the second equivalent method (described in the first paper (reviewed above)) is studied as a method for the development on new numerical techniques. A direct numerical correction of the truncation error terms in the second equivalent equation is introduced. The asymptotic successive-correction techniques are applied to the correction of the second equivalent equation and used to obtain higher-order numerical methods with good stability properties. The direct-correction and successive-correction numerical techniques are applied to some systems of ordinary differential equations and are compared with Runge-Kutta techniques.