A moving mesh numerical method for hyperbolic conservation laws
Author:
Bradley J. Lucier
Journal:
Math. Comp. 46 (1986), 5969
MSC:
Primary 65M25; Secondary 35L05
MathSciNet review:
815831
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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 STANCS82924 (dissertation).
 [2]
J. H. Bolstad, An Adaptive Finite Difference Method for Hyperbolic Systems in One Space Dimension, Lawrence Berkeley Lab. LBL13287 (STANCS82899) (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
(53 #6982)
 [4]
Michael
G. Crandall and Andrew
Majda, Monotone difference approximations for
scalar conservation laws, Math. Comp.
34 (1980), no. 149, 1–21. MR 551288
(81b:65079), http://dx.doi.org/10.1090/S00255718198005512883
 [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
(46 #2210)
 [6]
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
(83d:65259), http://dx.doi.org/10.1137/0903002
 [7]
Todd
Dupont, Mesh modification for evolution
equations, Math. Comp. 39
(1982), no. 159, 85–107. MR 658215
(84g:65131), http://dx.doi.org/10.1090/S00255718198206582150
 [8]
Ami
Harten, High resolution schemes for hyperbolic conservation
laws, J. Comput. Phys. 49 (1983), no. 3,
357–393. MR
701178 (84g:65115), http://dx.doi.org/10.1016/00219991(83)901365
 [9]
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
(85g:65111), http://dx.doi.org/10.1016/00219991(83)900669
 [10]
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 0413526
(54 #1640)
 [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
(50 #9169)
 [12]
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
(84c:65122)
 [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 (85k:65081), http://dx.doi.org/10.1016/03770427(83)900092
 [14]
S. N. Kružkov, "First order quasilinear equations with several space variables," Math. USSR Sb., v. 10, 1970, pp. 217243.
 [15]
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.
 [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
(50 #2709)
 [17]
Randall
J. LeVeque, Large time step shockcapturing techniques for scalar
conservation laws, SIAM J. Numer. Anal. 19 (1982),
no. 6, 1091–1109. MR 679654
(84e:65099), http://dx.doi.org/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
(86d:65123), http://dx.doi.org/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
(88a:65104), http://dx.doi.org/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
(87j:65110), http://dx.doi.org/10.1090/S00255718198608421216
 [21]
Keith
Miller and Robert
N. Miller, Moving finite elements. I, SIAM J. Numer. Anal.
18 (1981), no. 6, 1019–1032. MR 638996
(84m:65113a), http://dx.doi.org/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, BerlinNew York, 1979,
pp. 171–184. MR 540136
(80k:86003)
 [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
(85i:65121), http://dx.doi.org/10.1090/S00255718198307176898
 [25]
John
R. Rice, The approximation of functions. Vol. I: Linear
theory, AddisonWesley Publishing Co., Reading, Mass.London, 1964. MR 0166520
(29 #3795)
 [26]
Richard
Sanders, On convergence of monotone finite
difference schemes with variable spatial differencing, Math. Comp. 40 (1983), no. 161, 91–106. MR 679435
(84a:65075), http://dx.doi.org/10.1090/S00255718198306794356
 [27]
Richard
Sanders, The moving grid method for nonlinear hyperbolic
conservation laws, SIAM J. Numer. Anal. 22 (1985),
no. 4, 713–728. MR 795949
(87f:65110), http://dx.doi.org/10.1137/0722043
 [1]
 M. Berger, Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations, Stanford Computer Science Report STANCS82924 (dissertation).
 [2]
 J. H. Bolstad, An Adaptive Finite Difference Method for Hyperbolic Systems in One Space Dimension, Lawrence Berkeley Lab. LBL13287 (STANCS82899) (dissertation).
 [3]
 C. de Boor, "Good approximation by splines with variable knots," in Spline Functions and Approximation Theory (A. Meir and A. Sharma, eds.), ISNM, v. 21, BirkhäuserVerlag, Basel, 1973, pp. 5772. MR 0403169 (53:6982)
 [4]
 M. G. Crandall & A. Majda, "Monotone difference approximations for scalar conservation laws," Math. Comp., v. 34, 1980, pp. 121. MR 551288 (81b:65079)
 [5]
 C. M. Dafermos, "Polygonal approximations of solutions of the initial value problem for a conservation law," J. Math. Anal. Appl., v. 38, 1972, pp. 3341. MR 0303068 (46:2210)
 [6]
 S. F. Davis & J. E. Flaherty, "An adaptive finite element method for initialvalue problems for partial differential equations," SIAM J. Sci. Statist. Comput., v. 3, 1982, pp. 628. MR 651864 (83d:65259)
 [7]
 T. Dupont, "Mesh modification for evolution equations," Math. Comp., v. 39, 1982, pp. 85107. MR 658215 (84g:65131)
 [8]
 A. Harten, "High resolution schemes for hyperbolic conservation laws," J. Comput. Phys., v. 49, 1983, pp. 357393. MR 701178 (84g:65115)
 [9]
 A. Harten & J. M. Hyman, "Selfadjusting grid methods for onedimensional hyperbolic conservation laws," J. Comput. Phys., v. 50, 1983, pp. 235269. MR 707200 (85g:65111)
 [10]
 A. Harten, J. M. Hyman & P. D. Lax, "On finite difference approximations and entropy conditions for shocks," Comm. Pure Appl. Math., v. 29, 1976, pp. 297322. MR 0413526 (54:1640)
 [11]
 G. W. Hedstrom, "Some numerical experiments with Dafermos's method for nonlinear hyperbolic equations," Lecture Notes in Math., vol. 267, SpringerVerlag, Berlin and New York, 1972, pp. 117138. MR 0356699 (50:9169)
 [12]
 G. W. Hedstrom & G. H. Rodrigue, "Adaptivegrid methods for timedependent partial differential equations," in Multigrid Methods (W. Hackbusch and U. Trottenberg, eds.), SpringerVerlag, Berlin and New York, 1982, pp. 474484. MR 685784 (84c:65122)
 [13]
 B. M. Herbst, S. W. Schoombie & A. R. Mitchell, "Equidistributing principles in moving finite element methods," J. Comput. Appl. Math., v. 9, 1983, pp. 377489. MR 729241 (85k:65081)
 [14]
 S. N. Kružkov, "First order quasilinear equations with several space variables," Math. USSR Sb., v. 10, 1970, pp. 217243.
 [15]
 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.
 [16]
 P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM Regional Conference Lectures in Applied Mathematics, no. 11, 1972. MR 0350216 (50:2709)
 [17]
 R. LeVeque, "Large time step shockcapturing techniques for scalar conservation laws," SIAM J. Numer. Anal., v. 19, 1982, pp. 10911109. MR 679654 (84e:65099)
 [18]
 B. J. Lucier, "A stable adaptive numerical scheme for hyperbolic conservation laws," SIAM J. Numer. Anal., v. 22, 1985, pp. 180203. MR 772891 (86d:65123)
 [19]
 B. J. Lucier, "Error bounds for the methods of Glimm, Godunov, and LeVeque," SIAM J. Numer. Anal., Dec. 1985. MR 811184 (88a:65104)
 [20]
 B. J. Lucier, "On nonlocal monotone difference methods for scalar conservation laws," Math. Comp. (To appear.) MR 842121 (87j:65110)
 [21]
 K. Miller, "Moving finite elements, part II," SIAM J. Numer. Anal., v. 18, 1981, pp. 10191057. MR 638996 (84m:65113a)
 [22]
 J. Oliger, "Approximate methods for atmospheric and oceanographic circulation problems," Lecture Notes in Physics, vol. 91 (R. Glowinski and J. Lions, eds.), SpringerVerlag, Berlin and New York, 1979, pp. 171184. MR 540136 (80k:86003)
 [23]
 S. Osher & S. Chakravarthy, High Resolution Schemes and the Entropy Condition, ICASE Report 172218.
 [24]
 S. Osher & R. Sanders, "Numerical approximations to nonlinear conservation laws with locally varying time and space grids," Math. Comp., v. 41, 1983, pp. 321336. MR 717689 (85i:65121)
 [25]
 J. R. Rice, The Approximation of Functions, Vol. 1, AddisonWesley, Reading, Mass., 1964. MR 0166520 (29:3795)
 [26]
 R. Sanders, "On convergence of monotone finite difference schemes with variable spatial differencing," Math. Comp., v. 40, 1983, pp. 91106. MR 679435 (84a:65075)
 [27]
 R. Sanders, "The moving grid method for nonlinear hyperbolic conservation laws," SIAM J. Numer. Anal., v. 22, 1985, pp. 713728. MR 795949 (87f:65110)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608158314
PII:
S 00255718(1986)08158314
Keywords:
Conservation law,
adaptive methods,
method of characteristics
Article copyright:
© Copyright 1986
American Mathematical Society
