The Hermite polynomials are simple, effective interpolants of discrete data. These interpolants can preserve local positivity, monotonicity, and convexity of the data if we restrict their derivatives to satisfy constraints at the data points. This paper de- scribes the conditions that must be satisfied for cubic and quintic Hermite interpolants to preserve these properties when they exist in the discrete data. We construct algorithms to ensure that these constraints are satisfied and give numerical examples to illustrate the effectiveness of the algorithms on locally smooth and rough data. 1. Introduction. Piecewise polynomial interpolants, especially those based on Hermite polynomials (polynomials determined by their values and values of one or more derivatives at both ends of an interval), have a number of desirable properties. They are easy to compute once the derivative values are chosen. If the derivative values are chosen locally (e.g., by finite difference methods), then the interpolant at a given point will depend only on the given data at nearby mesh points. If the derivatives are computed by spline methods, then the interpolant will have an extra degree of continuity at the mesh points. In either case, the interpolant is linear in the given function values and has excellent convergence properties as the mesh spacing decreases. These methods, however, do not necessarily preserve the shape of the given data. When the data arise from a physical experiment, it may be vital that the interpolant preserve nonnegativity (f(x) > 0), nonpositivity (f(x) 0 or f(x) 0), or concavity (f(x) < 0). In this and other cases, geometric considerations, such as preventing spurious behavior near rapid changes in the data, may be more important than the asymptotic accuracy of the interpolation method. One can construct a shape-preserving interpolant by constraining the derivatives for the Hermite polynomials to meet conditions which imply the desired properties ((4), (5), (8), (11)—(15), (20)), by adding new mesh points

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Nonnegativity-, monotonicity-, or convexity-preserving cubic and quintic Hermite interpolation

Monotone piecewise cubic interpolants are simple and effective. They are generally third-order accurate, except near strict local extrema where accuracy degenerates to second-order due to the monotonicity constraint. Algorithms for piecewise cubic interpolants, which preserve monotonicity as well as uniform third and fourth-order accuracy are presented. The gain of accuracy is obtained by relaxing the monotonicity constraint in a geometric framework in which the median function plays a crucial role.

/pdf/accurate-monotone-cubic-interpolation-4xr9swlqsm.pdf

Accurate monotone cubic interpolation

For linear and fully non-linear diffusion equations of Bellman-Isaacs type, we introduce a class of approximation schemes based on differencing and interpolation. As opposed to classical numerical methods, these schemes work for general diffusions with coefficient matrices that may be non-diagonal dominant and arbitrarily degenerate. In general such schemes have to have a wide stencil. Besides providing a unifying framework for several known first order accurate schemes, our class of schemes includes new first and higher order versions. The methods are easy to implement and more efficient than some other known schemes. We prove consistency and stability of the methods, and for the monotone first order methods, we prove convergence in the general case and robust error estimates in the convex case. The methods are extensively tested.

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Semi-Lagrangian schemes for linear and fully non-linear diffusion equations

https://www-cs.ccny.cuny.edu/~wolberg/pub/jcam02.pdf

An energy-minimization framework for monotonic cubic spline interpolation

$C^1 $ monotone quadratic splines are analyzed. This motivates an algorithm for interpolating monotone data, given on a rectangular grid, with a $C^1 $ monotone quadratic spline surface. Error estimates, an operation count and numerical examples are given.

Monotonicity Preserving Surface Interpolation

Fritsch and Carlson [SIAM J. Numer. Anal., 17 (1980), pp. 238–246] developed an algorithm which produces a monotone $C^1 $ piecewise cubic interpolant to a monotone function. We show that their algorithm yields a third-order approximation, while a modification is fourth-order accurate.

S. C. Eisenstat

Papers

The Order of Monotone Piecewise Cubic Interpolation.