Essentially Nonoscillatory Postprocessing Filtering Methods

  • F. Lafon
  • S. Osher
Conference paper
Part of the ICASE/NASA LaRC Series book series (ICASE/NASA)


High order accurate centered flux approximations used in the computation of numerical solutions to nonlinear partial differential equations produce large oscillations in regions of sharp transitions. In this paper, we present a new class of filtering methods denoted by ENO-LS (Essentially Nonoscillatory Least Squares) which constructs an upgraded filtered solution that is close to the physically correct weak solution of the original evolution equation. Our method relies on the evaluation of a least squares polynomial approximation to oscillatory data using a set of points which is determined via the ENO framework.

Numerical results are given in one and two space dimensions for both scalar and systems of hyperbolic conversation laws. Computational running time, efficiency and robustness of the method are illustrated in various examples such as Riemann initial data for both Burger’s and Euler’s equations of gas dynamics. In all standard cases the filtered solution appears to converge numerically to the correct solution of the original problem. Some interesting results based on nonstandard central difference schemes, which exactly preserve entropy, and have been recently shown generally not to be weakly convergent to a solutiom of the conversation law, are also obtained using our filters.


Entropy Dinate 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    B. Engquist, P. Lotstedt, and B. Sjogreen, Math. Comput., 52, 509 (1989).MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, J. Comput. Phys., 71, 231 (1987).MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    J. Goodman and P.D. Lax, Comm. Pure. Appl. Math., 41, 591 (1988).MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    T. Hou and P.D. Lax, Comm. Pure. Appl. Math., 44, 1 (1991).MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    F. Lafon and S. Osher, “High order filtering methods for approximating hyperbolic systems of conservation laws,” ICASE Report, 90-25, 1990, J. Comput. Phys., to appear (1992).Google Scholar
  6. [6]
    D. Levermore and J. Liu, to appear.Google Scholar
  7. [7]
    C-W. Shu and S. Osher, J. Comput. Phys., 77, 439 (1988).MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    C-W. Shu and S. Osher, J. Comput. Phys., 83, 32 (1989).MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    J. Von Neumann and R.D. Richtmyer, J. Appl. Phys., 21, 380 (1950).CrossRefGoogle Scholar
  10. [10]
    T. Zang, M. Hussaini, and D. Bushnell, AIAA J., 22, 13 (1984).MATHCrossRefGoogle Scholar
  11. [11]
    T. Zang, M. Kopriva, and M. Hussaini, “Pseudospectral Calculation of Shock Turbulence Interactions,” ICASE Report 83 - 14, 1983.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • F. Lafon
    • 1
  • S. Osher
    • 2
  1. 1.C.E.A - CEL-VVilleneuve Saint GeorgesFrance
  2. 2.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

Personalised recommendations