Supra-convergent schemes on irregular grids

Authors:
H.-O. Kreiss, T. A. Manteuffel, B. Swartz, B. Wendroff and A. B. White

Journal:
Math. Comp. **47** (1986), 537-554

MSC:
Primary 65L05; Secondary 40A30, 65D25, 65L10

MathSciNet review:
856701

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Abstract | References | Similar Articles | Additional Information

Abstract: As Tikhonov and Samarskiĭ showed for , it is not essential that *k*th-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 . For any *k*, in fact, there is a -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 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 three-periodic 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.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0856701-5

Keywords:
Initial value problems,
boundary value problems,
compact difference schemes,
irregular grids,
nonuniform mesh,
difference quotients,
divided differences,
truncation error,
supra-convergence,
superconvergence

Article copyright:
© Copyright 1986
American Mathematical Society