A moving mesh numerical method for hyperbolic conservation laws
Author:
Bradley J. Lucier
Journal:
Math. Comp. 46 (1986), 5969
MSC:
Primary 65M25; Secondary 35L05
DOI:
https://doi.org/10.1090/S00255718198608158314
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 ${L^1}({\mathbf {R}})$ to within $O({N^{  2}})$ by a piecewise linear function with $O(N)$ 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 $O({N^{  1}})$. These numerical methods for conservation laws are the first to have proven convergence rates of greater than $O({N^{  1/2}})$.

M. Berger, Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations, Stanford Computer Science Report STANCS82924 (dissertation).
J. H. Bolstad, An Adaptive Finite Difference Method for Hyperbolic Systems in One Space Dimension, Lawrence Berkeley Lab. LBL13287 (STANCS82899) (dissertation).
 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
 Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1–21. MR 551288, DOI https://doi.org/10.1090/S00255718198005512883
 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 303068, DOI https://doi.org/10.1016/0022247X%2872%2990114X
 Stephen F. Davis and Joseph E. Flaherty, An adaptive finite element method for initialboundary value problems for partial differential equations, SIAM J. Sci. Statist. Comput. 3 (1982), no. 1, 6–27. MR 651864, DOI https://doi.org/10.1137/0903002
 Todd Dupont, Mesh modification for evolution equations, Math. Comp. 39 (1982), no. 159, 85–107. MR 658215, DOI https://doi.org/10.1090/S00255718198206582150
 Ami Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1983), no. 3, 357–393. MR 701178, DOI https://doi.org/10.1016/00219991%2883%29901365
 Ami Harten and James M. Hyman, Selfadjusting grid methods for onedimensional hyperbolic conservation laws, J. Comput. Phys. 50 (1983), no. 2, 235–269. MR 707200, DOI https://doi.org/10.1016/00219991%2883%29900669
 A. Harten, J. M. Hyman, and P. D. Lax, On finitedifference approximations and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), no. 3, 297–322. With an appendix by B. Keyfitz. MR 413526, DOI https://doi.org/10.1002/cpa.3160290305
 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
 G. W. Hedstrom and G. H. Rodrique, Adaptivegrid methods for timedependent partial differential equations, Multigrid methods (Cologne, 1981) Lecture Notes in Math., vol. 960, Springer, BerlinNew York, 1982, pp. 474–484. MR 685784
 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, DOI https://doi.org/10.1016/03770427%2883%29900092 S. N. Kružkov, "First order quasilinear equations with several space variables," Math. USSR Sb., v. 10, 1970, pp. 217243. N. N. Kuznetsov, "Accuracy of some approximate methods for computing the weak solutions of a firstorder quasilinear equation," USSR Comput. Math. and Math. Phys., v. 16, no. 6, 1976, pp. 105119.
 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
 Randall J. LeVeque, Large time step shockcapturing techniques for scalar conservation laws, SIAM J. Numer. Anal. 19 (1982), no. 6, 1091–1109. MR 679654, DOI https://doi.org/10.1137/0719080
 Bradley J. Lucier, A stable adaptive numerical scheme for hyperbolic conservation laws, SIAM J. Numer. Anal. 22 (1985), no. 1, 180–203. MR 772891, DOI https://doi.org/10.1137/0722012
 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, DOI https://doi.org/10.1137/0722064
 Bradley J. Lucier, On nonlocal monotone difference schemes for scalar conservation laws, Math. Comp. 47 (1986), no. 175, 19–36. MR 842121, DOI https://doi.org/10.1090/S00255718198608421216
 Keith Miller and Robert N. Miller, Moving finite elements. I, SIAM J. Numer. Anal. 18 (1981), no. 6, 1019–1032. MR 638996, DOI https://doi.org/10.1137/0718070
 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, BerlinNew York, 1979, pp. 171–184. MR 540136 S. Osher & S. Chakravarthy, High Resolution Schemes and the Entropy Condition, ICASE Report 172218.
 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, DOI https://doi.org/10.1090/S00255718198307176898
 John R. Rice, The approximation of functions. Vol. I: Linear theory, AddisonWesley Publishing Co., Reading, Mass.London, 1964. MR 0166520
 Richard Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), no. 161, 91–106. MR 679435, DOI https://doi.org/10.1090/S00255718198306794356
 Richard Sanders, The moving grid method for nonlinear hyperbolic conservation laws, SIAM J. Numer. Anal. 22 (1985), no. 4, 713–728. MR 795949, DOI https://doi.org/10.1137/0722043
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Additional Information
Keywords:
Conservation law,
adaptive methods,
method of characteristics
Article copyright:
© Copyright 1986
American Mathematical Society