A moving mesh numerical method for hyperbolic conservation laws

Author:
Bradley J. Lucier

Journal:
Math. Comp. **46** (1986), 59-69

MSC:
Primary 65M25; Secondary 35L05

MathSciNet review:
815831

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that the possibly discontinuous solution of a scalar conservation law in one space dimension may be approximated in to within by a piecewise linear function with nodes; the nodes are moved according to the method of characteristics. We also show that a previous method of Dafermos, which uses piecewise constant approximations, is accurate to . These numerical methods for conservation laws are the first to have proven convergence rates of greater than .

**[1]**M. Berger,*Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations*, Stanford Computer Science Report STAN-CS-82-924 (dissertation).**[2]**J. H. Bolstad,*An Adaptive Finite Difference Method for Hyperbolic Systems in One Space Dimension*, Lawrence Berkeley Lab. LBL-13287 (STAN-CS-82-899) (dissertation).**[3]**Carl de Boor,*Good approximation by splines with variable knots*, Spline functions and approximation theory (Proc. Sympos., Univ. Alberta, Edmonton, Alta., 1972) Birkhäuser, Basel, 1973, pp. 57–72. Internat. Ser. Numer. Math., Vol. 21. MR**0403169****[4]**Michael G. Crandall and Andrew Majda,*Monotone difference approximations for scalar conservation laws*, Math. Comp.**34**(1980), no. 149, 1–21. MR**551288**, 10.1090/S0025-5718-1980-0551288-3**[5]**Constantine M. Dafermos,*Polygonal approximations of solutions of the initial value problem for a conservation law*, J. Math. Anal. Appl.**38**(1972), 33–41. MR**0303068****[6]**Stephen F. Davis and Joseph E. Flaherty,*An adaptive finite element method for initial-boundary value problems for partial differential equations*, SIAM J. Sci. Statist. Comput.**3**(1982), no. 1, 6–27. MR**651864**, 10.1137/0903002**[7]**Todd Dupont,*Mesh modification for evolution equations*, Math. Comp.**39**(1982), no. 159, 85–107. MR**658215**, 10.1090/S0025-5718-1982-0658215-0**[8]**Ami Harten,*High resolution schemes for hyperbolic conservation laws*, J. Comput. Phys.**49**(1983), no. 3, 357–393. MR**701178**, 10.1016/0021-9991(83)90136-5**[9]**Ami Harten and James M. Hyman,*Self-adjusting grid methods for one-dimensional hyperbolic conservation laws*, J. Comput. Phys.**50**(1983), no. 2, 235–269. MR**707200**, 10.1016/0021-9991(83)90066-9**[10]**A. Harten, J. M. Hyman, and P. D. Lax,*On finite-difference approximations and entropy conditions for shocks*, Comm. Pure Appl. Math.**29**(1976), no. 3, 297–322. With an appendix by B. Keyfitz. MR**0413526****[11]**G. W. Hedstrom,*Some numerical experiments with Dafermos’s method for nonlinear hyperbolic equations*, Numerische Lösung nichtlinearer partieller Differential- und Integrodifferentialgleichungen (Tagung, Math. Forschungsinst., Oberwolfach, 1971), Springer, Berlin, 1972, pp. 117–138. Lecture Notes in Math., Vol. 267. MR**0356699****[12]**G. W. Hedstrom and G. H. Rodrique,*Adaptive-grid methods for time-dependent partial differential equations*, Multigrid methods (Cologne, 1981) Lecture Notes in Math., vol. 960, Springer, Berlin-New York, 1982, pp. 474–484. MR**685784****[13]**B. M. Herbst, S. W. Schoombie, and A. R. Mitchell,*Equidistributing principles in moving finite element methods*, J. Comput. Appl. Math.**9**(1983), no. 4, 377–389. MR**729241**, 10.1016/0377-0427(83)90009-2**[14]**S. N. Kružkov, "First order quasilinear equations with several space variables,"*Math. USSR Sb.*, v. 10, 1970, pp. 217-243.**[15]**N. N. Kuznetsov, "Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation,"*USSR Comput. Math. and Math. Phys.*, v. 16, no. 6, 1976, pp. 105-119.**[16]**Peter D. Lax,*Hyperbolic systems of conservation laws and the mathematical theory of shock waves*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. MR**0350216****[17]**Randall J. LeVeque,*Large time step shock-capturing techniques for scalar conservation laws*, SIAM J. Numer. Anal.**19**(1982), no. 6, 1091–1109. MR**679654**, 10.1137/0719080**[18]**Bradley J. Lucier,*A stable adaptive numerical scheme for hyperbolic conservation laws*, SIAM J. Numer. Anal.**22**(1985), no. 1, 180–203. MR**772891**, 10.1137/0722012**[19]**Bradley J. Lucier,*Error bounds for the methods of Glimm, Godunov and LeVeque*, SIAM J. Numer. Anal.**22**(1985), no. 6, 1074–1081. MR**811184**, 10.1137/0722064**[20]**Bradley J. Lucier,*On nonlocal monotone difference schemes for scalar conservation laws*, Math. Comp.**47**(1986), no. 175, 19–36. MR**842121**, 10.1090/S0025-5718-1986-0842121-6**[21]**Keith Miller and Robert N. Miller,*Moving finite elements. I*, SIAM J. Numer. Anal.**18**(1981), no. 6, 1019–1032. MR**638996**, 10.1137/0718070**[22]**Joseph Oliger,*Approximate methods for atmospheric and oceanographic circulation problems*, Computing methods in applied sciences and engineering (Proc. Third Internat. Sympos., Versailles, 1977) Lecture Notes in Phys., vol. 91, Springer, Berlin-New York, 1979, pp. 171–184. MR**540136****[23]**S. Osher & S. Chakravarthy,*High Resolution Schemes and the Entropy Condition*, ICASE Report 172218.**[24]**Stanley Osher and Richard Sanders,*Numerical approximations to nonlinear conservation laws with locally varying time and space grids*, Math. Comp.**41**(1983), no. 164, 321–336. MR**717689**, 10.1090/S0025-5718-1983-0717689-8**[25]**John R. Rice,*The approximation of functions. Vol. I: Linear theory*, Addison-Wesley Publishing Co., Reading, Mass.-London, 1964. MR**0166520****[26]**Richard Sanders,*On convergence of monotone finite difference schemes with variable spatial differencing*, Math. Comp.**40**(1983), no. 161, 91–106. MR**679435**, 10.1090/S0025-5718-1983-0679435-6**[27]**Richard Sanders,*The moving grid method for nonlinear hyperbolic conservation laws*, SIAM J. Numer. Anal.**22**(1985), no. 4, 713–728. MR**795949**, 10.1137/0722043

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0815831-4

Keywords:
Conservation law,
adaptive methods,
method of characteristics

Article copyright:
© Copyright 1986
American Mathematical Society