FINITE DIFFERENCES ASSOCIATED WITH SECOND—ORDER DIFFERENTIAL EQUATIONS

Abstract

Finite-difference approximations to second-order differential equations are generally less satisfactory when terms involving first derivatives are present. We here discuss some methods of approximation which are aimed at improving the accuracy in such cases. They arise out of a closer investigation of a method originally proposed by D.N. de G.Allen and R.V.Southwell for dealing with a partial differential equation which occurs in the field of viscous fluid motion. The paper falls into three fairly distincet parts. In the first we examine the basic approximation and we are then led to consider whether equally satisfactory results cannot be obtained in certain cases when first-derivative terins are not present. Both ordinary and partial differential equations are considered up to this point; we then make an extension of the basic theory of more especial interest in the case of ordinary differential equations.