A High-Order Difference Method for Differential Equations3"

This paper analyzes a high-accuracy approximation to the mth-order linear ordinary differential equation Mu = f. At mesh points, U is the estimate of u; and U satisfies MnU = Inf, where MnU is a linear combination of values of U at m + 1 stencil points (adjacent mesh points) and Inf is a linear combination of values of f at J auxiliary points, which are between the first and last stencil points. The coefficients of Mn, In are obtained "locally" by solving a small linear system for each group of stencil points in order to make the approximation exact on a linear space S of dimension L + 1. For separated two-point boundary value problems, C is the solution of an nby-n linear system with full bandwidth m + 1. For S a space of polynomials, existence and uniqueness are established, and the discretization error is 0(h ); the first m — 1 divided differences of U tend to those of u at this rate. For a general set of auxiliary points one has L = J + m; but special auxiliary points, which depend upon M and the stencil points, allow larger L, up to L = 2J + m. Comparison of operation counts for this method and five other common schemes shows that this method is among the most efficient for given convergence rate. A brief selection from extensive experiments is presented which supports the theoretical results and the practicality of the method.