[Paper Review] Compact Finite Differences and Cubic Splines

Robert M. Corless|arXiv (Cornell University)|May 19, 2018

Advanced Numerical Analysis Techniques25 references3 citations

TL;DR

This paper establishes a novel connection between fourth-order compact finite differences and cubic splines on uniform meshes, showing that the Padé-based compact finite difference scheme produces the exact derivatives required for twice-continuous differentiability in cubic splines. It introduces a new compact cubic interpolant with higher-order accuracy at boundaries and proves the underlying matrices are totally nonnegative and positive definite under uniform or mildly nonuniform meshes.

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 extsl{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.

Motivation & Objective

• To reveal a previously unnoticed mathematical connection between fourth-order compact finite difference schemes and cubic spline derivatives on uniform meshes.
• To develop a new spline-like interpolant—called the compact cubic interpolant—that achieves higher-order accuracy at mesh boundaries compared to standard cubic splines.
• To provide a unified framework for handling boundary conditions without requiring users to choose between natural, clamped, or not-a-knot conditions.
• To prove that the system matrices for compact cubic splines are positive definite and totally nonnegative when mesh widths have consistent sign.

Proposed method

• Uses contour integral and residue calculus to analytically derive and explain the equivalence between the compact finite difference formula and the derivative values required for cubic splines.
• Derives the compact finite difference scheme via Padé approximation, ensuring fourth-order accuracy for interior nodes.
• Introduces a modified four-node compact finite difference formula at boundaries to compute derivatives with $O(h^4)$ accuracy, even on mildly nonuniform meshes.
• Applies barycentric form for stable and efficient evaluation of the resulting piecewise cubic interpolant.
• Proves that the coefficient matrices are totally nonnegative and positive definite when mesh widths are all positive or all negative.
• Compares the new compact cubic interpolant with classical splines and pchip, emphasizing improved derivative accuracy at nodes and simplified boundary treatment.

Experimental results

Research questions

• RQ1 Why do fourth-order compact finite difference schemes on uniform meshes produce the exact derivative values required for cubic splines?
• RQ2 Can a new interpolant be constructed that maintains the smoothness of cubic splines while achieving higher-order accuracy at the boundaries?
• RQ3 How do the stability and conditioning of the compact cubic interpolant compare to classical spline methods under nonuniform meshes?
• RQ4 What algebraic properties (e.g., definiteness) do the system matrices of compact cubic splines possess?
• RQ5 Is the observed equivalence between compact finite differences and cubic splines a coincidence, or is there a deeper analytical explanation?

Key findings

• On a uniform mesh, the fourth-order compact finite difference scheme via Padé approximation produces derivatives identical to those required for cubic splines, confirming a precise mathematical equivalence.
• The proposed compact cubic interpolant achieves $O(h^4)$ accuracy for derivatives at the endpoints, improving upon the $O(h^3)$ accuracy of standard endpoint conditions in classical splines.
• The system matrices for compact cubic splines are proven to be positive definite and totally nonnegative when all mesh widths have the same sign, ensuring numerical stability.
• The connection between compact finite differences and cubic splines is explained via contour integrals and residues, offering a complex-analytic justification for an otherwise surprising numerical coincidence.
• An alternative explanation based on the decomposition of splines into pure cubics and $|x - x_0|^3$ terms also accounts for the equivalence, though the residue method is preferred for its broader theoretical implications.
• The method is implemented in MATLAB as `compactcubic.m`, providing a practical, stable, and user-friendly alternative to traditional spline construction with automatic boundary handling.