Generalized nonstandard finite differences and physical applications

Nonstandard finite differences can be used to construct exact algorithms to solve some differential equations of physical interest such as the wave equation and Schrödinger’s equation. Even where exact algorithms do not exist, nonstandard finite differences can greatly improve the accuracy of low-order finite-difference algorithms with a computational cost low compared to higher-order schemes or finer gridding. While nonstandard finite differences have been applied successfully to a variety 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 dimensions, give example algorithms, and discuss practical applications. © 1998 American Institute of Physics.