New Methods for the Derivation of Stable Difference Representations for Differential Equations

Finite difference representations of second order differential equations are derived which have exact solutions in certain simple situations. It is shown that these representations are superior to standard truncated Taylor representations by applying them to a certain “difficult” problem. The principles involved are extended to the treatment of mixed end point conditions, and improvements over the standard treatment of derivative end point conditions are demonstrated by example. The representations are then extended to partial differential equations, and major properties are indicated. A two dimensional viscous flow problem is solved to illustrate the application to partial differential equations.