Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Nonlinear filters for efficient shock computation

Authors: Björn Engquist, Per Lötstedt and Björn Sjögreen
Journal: Math. Comp. 52 (1989), 509-537
MSC: Primary 65M05; Secondary 35L65
MathSciNet review: 955750
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A new type of methods for the numerical approximation of hyperbolic conservation laws with discontinuous solution is introduced. The methods are based on standard finite difference schemes. The difference solution is processed with a nonlinear conservation form filter at every time level to eliminate spurious oscillations near shocks. It is proved that the filter can control the total variation of the solution and also produce sharp discrete shocks. The method is simpler and faster than many other high resolution schemes for shock calculations. Numerical examples in one and two space dimensions are presented.

References [Enhancements On Off] (What's this?)

  • [1] P. Colella, "Glimm's method for gas dynamics," SIAM J. Sci. Statist. Comput., v. 3, 1982, pp. 76-110. MR 651869 (83f:76069)
  • [2] D. Gottlieb, Spectral Methods for Compressible Flow Problems, Lecture Notes in Physics, No. 218 (Soubbaramayer and J. P. Boujot, eds.), Springer-Verlag, Berlin and New York, 1985, pp. 48-61. MR 793590 (86g:76015)
  • [3] A. Harten, "High resolution schemes for hyperbolic conservation laws," J. Comput. Phys., v. 49, 1983, pp. 357-393. MR 701178 (84g:65115)
  • [4] A. Harten & G. Zwas, "Switched numerical Shuman filters for shock calculations," J. Engrg. Math., v. 6, 1972, pp. 207-216.
  • [5] P. Lax & B. Wendroff, "Systems of conservation laws," Comm. Pure Appl. Math., v. 13, 1960, pp. 217-237. MR 0120774 (22:11523)
  • [6] S. Osher & S. Chakravarthy, "High resolution schemes and the entropy condition," SIAM J. Numer. Anal., v. 21, 1984, pp. 955-984. MR 760626 (86a:65086)
  • [7] S. Osher, A. Harten, B. Engquist & S. Chakravarthy, "Some results on uniformly high-order accurate essentially nonoscillatory schemes," J. Appl. Numer. Math., v. 2, 1986, pp. 347-377. MR 863993 (88g:65089)
  • [8] R. D. Richtmyer & K. W. Morton, Difference methods for Initial Value Problems, 2nd ed., Interscience, New York, 1967. MR 0220455 (36:3515)
  • [9] A. Rizzi & L.-E. Eriksson, "Computation of flow around wings based on the Euler equations," J. Fluid Mech., v. 148, 1984, p. 45-71.
  • [10] P. L. Roe, "Approximate Riemann solvers, parameter vectors, and difference schemes," J. Comput. Phys., v. 43, 1981, pp. 357-372. MR 640362 (82k:65055)
  • [11] G. A. Sod, " A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws," J. Comput. Phys., v. 27, 1978, pp. 1-31. MR 0495002 (58:13770)
  • [12] B. Van Leer, "Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method," J. Comput. Phys., v. 32, 1979, 101-136.
  • [13] P. R. Woodward & P. Colella, "The numerical simulation of two-dimensional fluid flow with strong shocks," J. Comput. Phys., v. 54, 1984, pp. 115-173. MR 748569 (85e:76004)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M05, 35L65

Retrieve articles in all journals with MSC: 65M05, 35L65

Additional Information

Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society