On the method of modified equations. II: Numerical techniques based on the equivalent equation for the Euler forward difference method (Q1294222)

The paper is the second 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 I and III see ibid. 103, No.~2-3, 111-139 and 179-212 (1999; reviewed above and below).] The validity of the modified equation based on the equivalent equation is studied as a method for the development of new numerical techniques. A direct numerical correction of the truncation terms that appear in the equivalent equation is introduced. In order to obtain stable numerical methods, a technique for the development of higher order, stable, successive-correction methods is presented. The technique is based on the truncation error terms in the equivalent equation of the original numerical method, in particular the Euler forward method. The results of asymptotic successive-correction techniques are compared with those obtained with Runge-Kutta methods for several systems of nonlinear ordinary differential equations.