GENERALIZED NONSTANDARD FINITE-DIFFERENCES AND PHYSICAL APPLICATIONS

Nonstandard finite differences can be used to construct exact algorith ms to solve some differential equations of physical interest such as t he wave equation and Schrodinger's equation. Even where exact algorith ms do not exist, nonstandard finite differences can greatly improve th e accuracy of low-order finite-difference algorithms with a computatio nal cost low compared to higher-order schemes or finer gridding. While nonstandard finite differences have been applied successfully to a va riety of one-dimensional problems, they cannot be directly extended to higher dimensions without modification. In this article we generalize the nonstandard finite-difference methodology to two and three dimens ions, give example algorithms, and discuss practical applications. (C) 1998 American Institute of Physics.