Nonlinear filters for efficient shock computation
Authors:
Björn Engquist, Per Lötstedt and Björn Sjögreen
Journal:
Math. Comp. 52 (1989), 509537
MSC:
Primary 65M05; Secondary 35L65
MathSciNet review:
955750
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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.
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 [3]
 A. Harten, "High resolution schemes for hyperbolic conservation laws," J. Comput. Phys., v. 49, 1983, pp. 357393. MR 701178 (84g:65115)
 [4]
 A. Harten & G. Zwas, "Switched numerical Shuman filters for shock calculations," J. Engrg. Math., v. 6, 1972, pp. 207216.
 [5]
 P. Lax & B. Wendroff, "Systems of conservation laws," Comm. Pure Appl. Math., v. 13, 1960, pp. 217237. MR 0120774 (22:11523)
 [6]
 S. Osher & S. Chakravarthy, "High resolution schemes and the entropy condition," SIAM J. Numer. Anal., v. 21, 1984, pp. 955984. MR 760626 (86a:65086)
 [7]
 S. Osher, A. Harten, B. Engquist & S. Chakravarthy, "Some results on uniformly highorder accurate essentially nonoscillatory schemes," J. Appl. Numer. Math., v. 2, 1986, pp. 347377. MR 863993 (88g:65089)
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 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. 4571.
 [10]
 P. L. Roe, "Approximate Riemann solvers, parameter vectors, and difference schemes," J. Comput. Phys., v. 43, 1981, pp. 357372. 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. 131. MR 0495002 (58:13770)
 [12]
 B. Van Leer, "Towards the ultimate conservative difference scheme. V. A secondorder sequel to Godunov's method," J. Comput. Phys., v. 32, 1979, 101136.
 [13]
 P. R. Woodward & P. Colella, "The numerical simulation of twodimensional fluid flow with strong shocks," J. Comput. Phys., v. 54, 1984, pp. 115173. MR 748569 (85e:76004)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909557509
PII:
S 00255718(1989)09557509
Article copyright:
© Copyright 1989
American Mathematical Society
