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Dimensional splitting with front tracking and adaptive grid refinement. (English) Zbl 0923.65061
For the numerical solution of scalar hyperbolic conservation laws in more than one dimension, dimensional splitting schemes have been widely used and analyzed. However, some effects due to the higher dimension may be not resolved, and spurious effects due to the grid chosen can occur. The authors prove error bounds for a second-order splitting scheme due to Strang using a front tracking method for approximating the one-dimensional solution operator. Numerical experiments show that the error bounds represent a worst-case scenario.
The influence of the CFL number is analyzed, too. In order to obtain higher accuracy in space, an adaptive grid refinement is included in the solution method.
Reviewer: E.Emmrich (Berlin)

MSC:
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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References:
[1] Godunov, Mat. Sbornik 47 pp 271– (1959)
[2] Strang, SIAM J. Numer. Anal. 5 pp 506– (1968)
[3] Crandall, Numer. Math. 34 pp 285– (1980)
[4] Teng, SIAM J. Numer. Anal. 31 pp 43– (1994)
[5] Holden, Comput. Math. Applic. 15 pp 595– (1988)
[6] Holden, Math. Comp. 60 pp 221– (1993)
[7] Roe, Numer. Meth. for PDEs 7 pp 277– (1991) · Zbl 0741.65077 · doi:10.1002/num.1690070306
[8] LeVeque, SIAM J. Numer. Anal. 19 pp 1091– (1982)
[9] Dafermos, J. Math. Anal. Appl. 38 pp 33– (1972)
[10] Langseth, Advances in Engineering Software. 26 pp 45– (1996)
[11] Kuznetsov, USSR Comput. Math. and Math. Phys. Dokl. 16 pp 105– (1976)
[12] Lucier, SIAM J. Numer. Anal. 22 pp 1074– (1985)
[13] Lucier, Math. Comp. 46 pp 59– (1986)
[14] Cockburn, Math. Comp. 65 pp 533– (1996)
[15] Kružkov, Math. USSR Sbornik 10 pp 217– (1970)
[16] Bratvedt, Surv. Math. Indu. 3 pp 185– (1993)
[17] Risebro, SIAM J. Sci. Stat. Comput. 12 pp 1401– (1991)
[18] Risebro, J. Comp. Phys. 101 pp 130– (1992)
[19] CLAWPACK User Notes, Applied Mathematics, Box 352420, University of Washington, Seattle, WA 98195-2420, Nov. 1995. Available from www.netlib.att.com in www.netlib/pdes/claw/doc/ or at the http://www.amath.washington.edu/rjl/clawpack.html.
[20] Harten, J. Comp. Phys. 50 pp 235– (1983)
[21] Karlsen, Comput. Methods Appl. Mech. Engrg.
[22] Karlsen, In Situ. 22 pp 59– (1998)
[23] Gropp, SIAM J. Sci. Stat. Comput. 8 pp 292– (1987)
[24] , and , ”A conservative front tracking scheme for 1D hyperbolic conservation laws.” In Proc. 4th Internat. Conf. Hyperbolic Problems, Taormina-Italy, 1992, pp. 385-392. · Zbl 0964.65517
[25] Risebro, Proc. Amer. Math. Soc. 117 pp 1125– (1993)
[26] Woodward, J. Comp. Phys. 54 pp 115– (1984)
[27] , and , ”An unconditionally stable method for the Euler equations,” to appear. · Zbl 0922.76251
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