Nonlinear filters for efficient shock computation

Engquist, Björn; Lötstedt, Per; Sjögreen, Björn

doi:10.1090/S0025-5718-1989-0955750-9

Nonlinear filters for efficient shock computation
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by Björn Engquist, Per Lötstedt and Björn Sjögreen PDF

Math. Comp. 52 (1989), 509-537 Request permission

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

Phillip Colella, Glimm’s method for gas dynamics, SIAM J. Sci. Statist. Comput. 3 (1982), no. 1, 76–110. MR 651869, DOI 10.1137/0903007
David Gottlieb, Spectral methods for compressible flow problems, Ninth international conference on numerical methods in fluid dynamics (Saclay, 1984) Lecture Notes in Phys., vol. 218, Springer, Berlin, 1985, pp. 48–61. MR 793590, DOI 10.1007/3-540-13917-6_{1}09
Ami Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1983), no. 3, 357–393. MR 701178, DOI 10.1016/0021-9991(83)90136-5

J. Engrg. Math.

Peter Lax and Burton Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217–237. MR 120774, DOI 10.1002/cpa.3160130205
Stanley Osher and Sukumar Chakravarthy, High resolution schemes and the entropy condition, SIAM J. Numer. Anal. 21 (1984), no. 5, 955–984. MR 760626, DOI 10.1137/0721060
Ami Harten, Stanley Osher, Björn Engquist, and Sukumar R. Chakravarthy, Some results on uniformly high-order accurate essentially nonoscillatory schemes, Appl. Numer. Math. 2 (1986), no. 3-5, 347–377. MR 863993, DOI 10.1016/0168-9274(86)90039-5
Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455

J. Fluid Mech.

P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43 (1981), no. 2, 357–372. MR 640362, DOI 10.1016/0021-9991(81)90128-5
Gary A. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys. 27 (1978), no. 1, 1–31. MR 495002, DOI 10.1016/0021-9991(78)90023-2

J. Comput. Phys.

Paul Woodward and Phillip Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys. 54 (1984), no. 1, 115–173. MR 748569, DOI 10.1016/0021-9991(84)90142-6