Symbolic Generation of Finite Difference Formulas

Tables of coefficients for high order accurate, compact approximations to the first ten derivatives on and at the midpoints of uniform nets are presented.The exact rational weights are generated and tested by means of symbolic manipulation implemented through MACSYMA.These weights are required in the application of deferred corrections to new methods for solving higher order two point boundary value problems.1. Introduction.Compact difference schemes are, by definition, those which use the least number of net points to obtain consistent approximations (i.e. at least first order accurate).Extending this definition, higher order compact schemes are those which use the least number of net points to obtain higher order accurate approximations.In this paper we derive and present tables of the coefficients for higher order compact approximations, from accuracy h2 to h10, to the first 10 derivatives of smooth functions on uniform nets.These approximations are particularly useful in applications of deferred corrections, and this was part of the motivation for the present work [5].Fast weight generators have also.beenused for this purpose [1], [2] but the current tables are more efficient.In addition, our tables are motivated by new schemes for solving higher order O.D.E.boundary value problems [5].The derivation of the formulas is classical, involving no more than Taylor expansions.However, the work is tedious and very prone to errors.Thus, we have used symbolic manipulation, implemented through MACSYMA, [6], both to derive the coefficients and to independently check their correctness.The coefficients are rational numbers and are given in their exact form as quotients of integers.Partial tables of some such coefficients have been published in [1], [7], [8], [9].The suggestion to use MACSYMA was made by the Numerical Analysis Group at Stanford University.The second author would like to thank Professor Gene H. Golub for his hospitality while visiting Stanford during the summer of 1975, where part of this work was carried out.2. Preliminaries.Given a uniform net {r} with step size h and a smooth function f(t) we study compact approximations to the derivatives of f(t) at t = 0 and at t = A/2, using only function values {f(t¡)}.We prefer compact approximations since they have truncation error expansions in powers of h2, they use a minimum number of ordinäres for a given order of accuracy and have the smallest weights and error constants.We use the notation Dpf(9) = dpf(9)/dtp, 0 = 0 or A/2, and EP2^f for the