Generalized finite-difference approximations f or the parallel solution of initial value problems

Abstract

Methods used in analog computation for the parallel- finite-differences solution of partial differential equations have been almost universally based on the "classical" derivation of finite-difference approximations. That is, the space-dependence of the approximate solution is lo cally assumed to be appropriately represented by a Tay lor (or polynomial) truncated series of low1, or higher7, order. Although known to introduce higher truncation errors than the use of other functional series2,3, this re striction has been mostly accepted for reasons of con venience because no simple method for obtaining more general approximations was available in practice. We show in this paper that completely general finite- difference approximations of linear partial differential operators in space, based on functional (rather than, but including, polynomial) approximations, can be easily ob tained in a direct method involving only the inversion and multiplication of constant matrices. One of the drawbacks of the analog computer imple mentation of equations obtained by application of the method presented here is that it requires more attenu ators than the classical low-order method. However, as suggested by Deiters and Nomura3, this is alleviated in a hybrid computer by introducing at discrete time inter vals the higher-order terms as digital corrections super imposed upon an analog low-order implementation. Examples of the derivation of generalized finite-differ ence equations are shown for the representation of the Laplacian (diffusion) operator in circular (r, θ) and linear (x) coordinates.