On the method of modified equations. I: Asymptotic analysis of the Euler forward difference method

The method of modi®ed equations is studied as a technique for the analysis of ®nite di€erence equations. The non-uniqueness of the modi®ed equation of a di€erence method is stressed and three kinds of modi®ed equations are introduced. The ®rst modi®ed or equivalent equation is the natural pseudo-di€erential operator associated to the original numerical method. Linear and nonlinear combinations of the equivalent equation and their derivatives yield the second modi®ed or second equivalent equation and the third modi®ed or (simply) modi®ed equation, respectively. For linear problems with constant coecients, the three kinds of modi®ed equations are equivalent among them and to the original di€erence scheme. For nonlinear problems, the three kinds of modi®ed equations are asymptotically equivalent in the sense that an asymptotic analysis of these equations with the time step as small parameter yields exactly the same results. In this paper, both regular and multiple scales asymptotic techniques are used for the analysis of the Euler forward di€erence method, and the resulting asymptotic expansions are veri®ed for several nonlinear, autonomous, ordinary di€erential equations. It is shown that, when the resulting asymptotic expansion is uniformly valid, the asymptotic method yields very accurate results if the solution of the leading order equation is smooth and does not blow up, even for large step sizes. Ó 1999 Elsevier Science Inc. All rights reserved.