Symbolic Generation of Finite Difference Formulas

Abstract

Abstract. 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 exactrational 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

Stencils with isotropic discretization error for differential operators

“…, x N ) that depends only on x 1 . Application of S then results in a one-dimensional finite-difference scheme for which it is known that its coefficients have alternating sign [3]. For r > 1, this violates the assumption of diagonal dominance as there all coefficients (except for the one in the centre) have the same sign [cf.…”

Generation of Finite Difference Formulas on Arbitrarily Spaced Grids

“…The most ambitious attempt to tabulate weights for many orders of derivatives and to high orders of accuracy appears to be the work by Keller and Pereyra [4]. However, their algorithms (limited to equispaced grids) were very involved, and the resulting tables contain both isolated and systematic errors.…”

Deferred corrections using uncentered end formulas

“…The coefficients ctij in (12) For the practical computation of the coefficients c%, see [9]. .jeS2 (20) Proof Recalling the definition of C k in (12), result (19a) follows from (16).…”