Supra-convergent schemes on irregular grids

Abstract:

As Tikhonov and Samarskiĭ showed for $k = 2$, it is not essential that kth-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 kth difference quotient is set equal to the value of the kth derivative at the middle point of the stencil; the proof is particularly transparent for $k = 2$. For any k, in fact, there is a $\left \lfloor {k/2} \right \rfloor$-parameter family of symmetric averages of the values of the kth 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 ${D^2}y = f$ 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.