Supra-Convergent Schemes on Irregular Grids

As Tikhonov and Samarskiï showed for k = 2, it is not essential that Ath-order compact difference schemes be centered at the arithmetic mean of the stencil's points to yield second-order convergence (although it does suffice). For stable schemes and even k, the main point is seen when the k th difference quotient is set equal to the value of the k th derivative at the middle point of the stencil; the proof is particularly transparent for k = 2. For any k, in fact, there is a [ k/2\ -parameter family of symmetric averages of the values of the k th derivative at the points of the stencil which, when similarly used, yield second-order convergence. The result extends to stable compact schemes for equations with lower-order terms under general boundary conditions. Although the extension of Numerov's tridiagonal scheme (approximating D2y = f with third-order truncation error) yields fourth-order convergence on meshes consisting of a bounded number of pieces in which the mesh size changes monotonically, it yields only third-order convergence to quintic polynomials on any threeperiodic mesh with unequal adjacent mesh sizes and fixed adjacent mesh ratios. A result of some independent interest is appended (and applied): it characterizes, simply, those functions of k variables which possess the property that their average value, as one translates over one period of an arbitrary periodic sequence of arguments, is zero; i.e., those bounded functions whose average value, as one translates over arbitrary finite sequences of arguments, goes to zero as the length of the sequences increases. 1. Some Supra-Convergent Schemes. The ordinary differential equation Dky = / with initial conditions Dpy = bp at x = 0, p = 0,1,..., k — 1, can be approximated by the finite-difference equation AkY = F with appropriate initial conditions on Y, where A* is the &th-order difference quotient, i.e., k\ times the divided difference on k + 1 points. For even k and a uniform grid with spacing h, as long as F is within 0(h2) of / at the middle point of the stencil, the truncation error Aky F is 0(h2), and so is the solution error Ap(y — Y), p < k. For odd k or nonuniform grids with maximum interval h, the truncation error may or may not be 0(h2). Nevertheless, we will show that for a class of F's the solution error remains 0(h2). We call such enhancement of truncation error sw/>ra-convergence. Received January 31,1984; revised April 9,1985. 1980 Mathematics Subject Classification. Primary 65L05, 65L10; Secondary 65D25, 40A30.