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On High-Order Accurate Interpolation for Non-Oscillatory Shock Capturing Schemes

  • Ami Harten
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 2)

Abstract

In this paper we describe high-order accurate Godunov-type schemes for the computation of weak solutions of hyperbolic conservation laws that are essentially non-oscillatory. We show that the problem of designing such schemes reduces to a problem in approximation of functions, namely that of reconstructing a piecewise smooth function from its given cell averages to high order accuracy and without introducing large spurious oscillatons. To solve this reconstruction problem we introduce a new interpolation technique that when applied to piecewise smooth data gives high-order accuracy wherever the function is smooth but avoids having a Gibbs-phenomenon at discontinuities.

Keywords

Truncation Error Local Extremum Interpolation Technique Entropy Solution Reconstruction Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    P. Colella and P.R. Woodward, The piecewise-parabolic method (PPM) for gas-dynamical simulations, J. Comp. Phys. v. 54, (1984), 174–201.MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. [2]
    A. Harten, J.M. Hyman and P.D. Lax (with appendix by B. Keyfitz), “On finite-differnce approximations and entropy conditions for shocks”, Comm. Pure Appl. Math., v. 29, (1976), 297–322.MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. [3]
    A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comp. Phys., 49 (1983), 357–393.MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. [4]
    A. Harten, On a class of high resolution total-variation-stable finite-difference schemes, SINUM, v. 21, (1984), 1–23.MathSciNetzbMATHGoogle Scholar
  5. [5]
    A. Harten and S. Osher, “Uniformly high-order accurate non-oscillatory schemes, I.”, MRC Technical Summary Report #2823, May 1985.Google Scholar
  6. [6]
    A. Harten, S. Osher, B. Engquist and S. Chakravarthy, “Uniformly high-order accurate non-oscillatory schemes, II”, in preparation.Google Scholar
  7. [7]
    S. Osher and S.R. Chakravarthy, “High-resolution schemes and the entropy condition”, SINUM, v. 21, (1984), 955–984.MathSciNetzbMATHGoogle Scholar
  8. [8]
    S. Osher and S.R. Chakravarthy, “Very high order accurate TVD schemes”, ICASE Report #84–44, (1984).Google Scholar
  9. [9]
    B. Van Leer, Towards the ultimate conservative scheme, II. Monotonicity and conservation combined in a second order scheme, J. Comp. Phys. 14 (1974), 361–376.ADSzbMATHCrossRefGoogle Scholar
  10. [10]
    P. Woodward, in Proceedings of the NATO Advanced Workshop in Astrophysical Radiation Hydrodynamics, Munich, West Germany, August 1982; also Lawrence Livermore Lab. Report #90009.Google Scholar
  11. [11]
    S.T. Zalesak, “Very high-order and pseudo-spectral flux-corrected transport (FCT) algorithms for conservation laws”, in “Advances in computer methods for partial differential equations”, Vol. 4 (R. Vichnevetsky and R.S. Stepleman, eds.) IMACS, Rutgers University, 1981.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Ami Harten
    • 1
    • 2
  1. 1.School of Mathematical SciencesTel-Aviv UniversityIsrael
  2. 2.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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