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arXiv · 1805.07659

Compact Finite Differences and Cubic Splines

Robert M. Corless
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Abstract

In this paper I uncover and explain---using contour integrals and residues---a connection between cubic splines and a popular compact finite difference formula. The connection is that on a uniform mesh the simplest Padé scheme for generating fourth-order accurate compact finite differences gives \textsl{exactly} the derivatives at the interior nodes needed to guarantee twice-continuous differentiability for cubic splines. %I found this connection surprising, because the two problems being solved are different. I also introduce an apparently new spline-like interpolant that I call a compact cubic interpolant; this is similar to one introduced in 1972 by Swartz and Varga, but has higher order accuracy at the edges. I argue that for mildly nonuniform meshes the compact cubic approach offers some potential advantages, and even for uniform meshes offers a simple way to treat the edge conditions, relieving the user of the burden of deciding to use one of the three standard options: free (natural), complete (clamped), or "not-a-knot" conditions. Finally, I establish that the matrices defining the compact cubic splines (equivalently, the fourth-order compact finite difference formulæ) are positive definite, and in fact totally nonnegative, if all mesh widths are the same sign.

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Robert M. Corless. 2019-11-22. Compact Finite Differences and Cubic Splines. https://arxiv.org/abs/1805.07659

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