A Reliable Method for the Numerical Integration of a Large Class of Ordinary and Partial Differential Equations

The purpose of this note is to describe the simple ex-tension of a popular method of solving second-order ordinary differential equations with two end-point bound-ary conditions to nth order ordinary differential equations and to partial differential equations that are second order in one direction. The orginal method is simply a version of Gaussian elimination; and the extension (to be de-scribed) has, we have discovered, been published, sliihtly differently, before. We feel, however, that the present note will be of value, since the extension has proven very useful to both of us, and is seldom used among numerical analysts and meteorologists. We begin by reviewing second-order ordinary differ-ential equations. Consider 8x is the grid interval used in finite-difference approxima-tion to equation (1), and A b f o + B b f, = D b, A l. f N- l + B l f N = D l where N is the level number corresponding to x = 1. The solution of equation (2) (following Richtmyer, 1957) goes as follows: f n = a n j n + l + P n (3) where an and)Bn are newly introduced variables. Then j n- l = a n- I j n S 8 n- 1. (4) Substituting equation (4) into (2) we obtain and Thus, knowing ao, bo we may readily obtain all a,'s and P i s. From the lower boundary condition dx2 %+g(x) g+h(x) j=r ( z) (7) where and df+alj=bl a t x = o dx and Equation (3) may be used to obtain j a t all n's, provided conditionwe have (9) (10) we know j N. With the upper boundary A t j N-- 1 + B t f N = D t. We also have