Supra-Convergent Schemes on Irregular Grids

• Published in: Mathematics of computation 1986, Vol.47 (176), p.537-554
• Main Authors: H.-O. Kreiss, Manteuffel, T. A., Swartz, B., Wendroff, B., White, A. B.
• Format: Article
• Language: English
• Subjects: Approximation; Boundary conditions; Boundary value problems; Difference quotients; Error rates; Exact sciences and technology; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Polynomials; Recursion; Sciences and techniques of general use; Stencils; Truncation errors
• ISSN: 0025-5718; 1088-6842
• DOI: 10.1090/s0025-5718-1986-0856701-5

As Tikhonov and Samarskiĭ showed for $k = 2$, 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 $k = 2$. For any $k$, in fact, there is a $\lfloor k/2\rfloor$-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 $D^2y = f$ 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 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.