On Shape-Preserving Interpolation and Semi-Lagrangian Transport

Abstract

A large number of interpolation schemes are evaluated in terms of their relative accuracy. The large number of schemes arises by considering combinations of interpolating forms (piecewise cubic polynomials, piecewise rational quadratic and cubic polynomials, and piecewise quadratic Bernstein polynomials), derivative estimates (Akima, Hyman, arithmetic, geometric and harmonic means, and Fritsch–Butland), and modification of these estimates required to ensure monotonicity and/or convexity upon the interpolant. Shape-preserving methods maintain in the interpolant the monotonicity and/or convexity implied in the discrete data.The schemes are first compared by evaluating their ability to interpolate evenly spaced data drawn from three test shapes (Gaussian, cosine bell, and triangle) at two resolutions. Details of the cosine bell tests are presented in this paper. Details of the other tests are presented in a companion technical report. Of the monotonic interpolants, the following are the most accurate: (1) The Hermite cubic interpolant with the derivative estimate of Hyman modified to produce monotonicity as suggested by de Boor and Swartz. (2) The second version of the rational cubic spline suggested by Delbourgo and Gregory, with the derivative estimate of Hyman modified to produce monotonicity. (3) The piecewise quadratic Bernstein polynomials suggested by McAllistor and Roulier with the derivative estimate of Hyman again modified. Imposing strict monotonicity at discrete extrema introduces significant errors. More accurate interpolations result if this requirement is relaxed at extrema. The Hermite cubic interpolant is improved by relaxing the strict monotonicity constraint to one suggested by Hyman at extrema. In a like manner, the accuracy of the rational and piecewise quadratic Bernstein polynomial interpolants can be improved by requiring only that convexity/concavity be satisfied rather than monotonicity.Some of the more accurate interpolants are incorporated into the semi-Lagrangian transport method and tested by examining the accuracy of the solution to one-dimensional advection of test shapes in a uniform velocity field. The semi-Lagrangian method using monotonic interpolators provides monotonic solutions. The semi-Lagrangian method using interpolators that maintain convex/concave constraints give solutions that are essentially nonoscillatory. The monotonic forms damp the solution with time, more so for narrow than broad structures. The best monotonic forms are the Hermite cubic interpolant with the Akima or Hyman derivative estimates modified to produce monotonicity with $C^0 $ continuity. The corresponding $C^1 $ continuous forms have unacceptable phase errors with the Hermite interpolant. The rational cubic with the Hyman derivative estimate modified to produce monotonicity is comparable to the $C^0 $ Hermite form described above. The $C^1 $ rational form does not have the phase error seen in the $C^1 $ Hermite interpolant. The essentially nonoscillatory forms damp much less than the monotonic forms. The solutions that used rational cubic interpolants with a Hyman derivative estimate modified to satisfy a convexity/concavity constraint were the most satisfactory of the shape-preserving schemes.