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

摘要

The method of modified equations is studied as a technique for the analysis of finite difference equations. The non-uniqueness of the modified equation of a difference method is stressed and three kinds of modified equations are introduced. The first modified or equivalent equation is the natural pseudo-differential operator associated to the original numerical method. Linear and nonlinear combinations of the equivalent equation and their derivatives yield the second modified or second equivalent equation and the third modified or (simply) modified equation, respectively. For linear problems with constant coefficients, the three kinds of modified equations are equivalent among them and to the original difference scheme. For nonlinear problems, the three kinds of modified 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 difference method, and the resulting asymptotic expansions are verified for several nonlinear, autonomous, ordinary differential 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.