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An Unconditionally Monotone <i>C</i><sup>2</sup> Quartic Spline Method with Nonoscillation Derivatives
Lawrence Livermore National Laboratory, Livermore, CA, USA
Lawrence Livermore National Laboratory, Livermore, CA, USA
- 1 Lawrence Livermore National Laboratory, Livermore, CA, USA
- 2 Lawrence Livermore National Laboratory, Livermore, CA, USA
Advances in Pure Mathematics·Volume 08 (2018)·Pages 25–40·Published 9 January 2018·DOI10.4236/apm.2018.81003
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Abstract
A one-dimensional monotone interpolation method based on interface reconstruction with partial volumes in the slope-space utilizing the Hermite cubic-spline, is proposed. The new method is only quartic, however is C 2 and unconditionally monotone. A set of control points is employed to constrain the curvature of the interpolation function and to eliminate possible nonphysical oscillations in the slope space. An extension of this method in two-dimensions is also discussed.
KeywordsMonotone InterpolationQuarticNon-Oscillation DerivativeInterface ReconstructionSlope SpaceHermit Spline
- Fritsch, F.N. and Carlson, R.E. (1980) Monotone Piecewise Cubic Interpolation. SIAM Journal on Numerical Analysis, 17, 238-246. https://doi.org/10.1137/0717021
- Fritsch, F.N. and Butland, J. (1984) A Method for Constructing Local Monotone Piecewise Cubic Interpolants. SIAM Journal on Scientific and Statistical Computing, 5, 300-304. http://epubs.siam.org/doi/abs/10.1137/0905021
- Carlson, R.E. and Fritsch, F.N. (1985) Monotone Piecewise Bicubic Interpolation. SIAM Journal on Numerical Analysis, 22, 386-400. https://doi.org/10.1137/0722023
- Steffen, M. (1990) A Simple Method for Monotonic Interpolation in One Dimension. Astronomy and Astrophysics, 239, 443.
- Heß, W. and Schmidt, J.W. (1994) Positive Quartic, Monotone Quintic C2-Spline Interpolation in One and Two Dimensions. Journal of Computational and Applied Mathematics, 55, 51-67. https://doi.org/10.1016/0377-0427(94)90184-8
- Ling, N.S., Kong, V.P. and Ong, B.H. (2009) A Class of Bézier-Like Splines in Smooth Monotone Interpolation. Malaysian Journal of Mathematical Sciences, 3, 203-226.
- Dougherty, R.L., Edelman, A. and Hyman, J.M. (1989) Nonnegativity-, Monotonicity-, or Convexity-Preserving Cubic and Quintic Hermite Interpolation. Mathematics of Computation (United States), 52, 471-494. https://doi.org/10.1090/S0025-5718-1989-0962209-1
- Jochen, W., Schmidt, J.W. and Heß, W. (1995) An Always Successful Method in Univariate Convex C2 Interpolation. Numerische Mathematik, 71, 237-252. https://doi.org/10.1007/s002110050143
- Sablonniere, P. and Mani, C. (1997) Monotone Interpolation of Order 3 by C2 Cubic Splines. IMA Journal of Numerical Analysis, 17, 305-320. https://doi.org/10.1093/imanum/17.2.305
- Sablonniere, P. and Merrien, J.-L. (2003) Monotone and Convex C1 Hermite Interpolants Generated by a Subdivision Scheme. Constructive Approximation, 19, 279-298. https://doi.org/10.1007/s00365-002-0512-3
- Hyman, J.M. (1983) Accurate Monotonicity Preserving Cubic Interpolation. SIAM Journal on Scientific and Statistical Computing, 4, 645-654. https://doi.org/10.1137/0904045
- Wolberg, G. and Alfy, I. (1999) Monotonic Cubic Spline Interpolation. Proceeding CGI’99 Proceedings of the International Conference on Computer Graphics, Washington DC, 188.
- Han, H. and Guo, X. (2018) Cubic Hermite Interpolation with Minimal Derivative Oscillation. Journal of Computational and Applied Mathematics, 331, 82-87. https://doi.org/10.1016/j.cam.2017.09.049