TVB RungeKutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework
Authors:
Bernardo Cockburn and ChiWang Shu
Journal:
Math. Comp. 52 (1989), 411435
MSC:
Primary 65M60; Secondary 35L65, 65N30
MathSciNet review:
983311
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Abstract: This is the second paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws . In this paper we present a general framework of the methods, up to any order of formal accuracy, using scalar onedimensional initial value and initialboundary problems as models. In these cases we prove TVBM (total variation bounded in the means), TVB, and convergence of the schemes. Numerical results using these methods are also given. Extensions to systems and/or higher dimensions will appear in future papers.
 [1]
A.
Bourgeat and Bernardo
Cockburn, A total variation diminishingprojection method for
solving implicit numerical schemes for scalar conservation laws: a
numerical study of a simple case, SIAM J. Sci. Statist. Comput.
10 (1989), no. 2, 253–273. MR 982223
(90g:65116), http://dx.doi.org/10.1137/0910018
 [2]
G. Chavent &, B. Cockburn, The local projection discontinuousGalerkin finite element method for scalar conservation laws, IMA Preprint Series #341, University of Minnesota, September 1987, to appear in .
 [3]
G.
Chavent and G.
Salzano, A finiteelement method for the 1D water flooding problem
with gravity, J. Comput. Phys. 45 (1982), no. 3,
307–344. MR
666166 (83g:76089), http://dx.doi.org/10.1016/00219991(82)901073
 [4]
B. Cockburn, The quasimonotone schemes for scalar conservation laws, I, II, and III, IMA Preprint Series #263, 268, 277, University of Minnesota, September and October 1986.
 [5]
B. Cockburn & C.W. Shu, The RungeKutta local projection discontinuousGalerkin finite element method for scalar conservation laws, IMA Preprint Series #388, University of Minnesota, 1988.
 [6]
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
 [7]
Ami
Harten, On a class of high resolution totalvariationstable
finitedifference schemes, SIAM J. Numer. Anal. 21
(1984), no. 1, 1–23. With an appendix by Peter D. Lax. MR 731210
(85f:65085), http://dx.doi.org/10.1137/0721001
 [8]
Ami
Harten, Preliminary results on the extension of ENO schemes to
twodimensional problems, Nonlinear hyperbolic problems (St.\ Etienne,
1986) Lecture Notes in Math., vol. 1270, Springer, Berlin, 1987,
pp. 23–40. MR 910102
(88k:65085), http://dx.doi.org/10.1007/BFb0078315
 [9]
Ami
Harten and Stanley
Osher, Uniformly highorder accurate nonoscillatory schemes.
I, SIAM J. Numer. Anal. 24 (1987), no. 2,
279–309. MR
881365 (90a:65198), http://dx.doi.org/10.1137/0724022
 [10]
Ami
Harten, Björn
Engquist, Stanley
Osher, and Sukumar
R. Chakravarthy, Uniformly highorder accurate essentially
nonoscillatory schemes. III, J. Comput. Phys. 71
(1987), no. 2, 231–303. MR 897244
(90a:65199), http://dx.doi.org/10.1016/00219991(87)900313
 [11]
Alexander
N. Brooks and Thomas
J. R. Hughes, Streamline upwind/PetrovGalerkin formulations for
convection dominated flows with particular emphasis on the incompressible
NavierStokes equations, Comput. Methods Appl. Mech. Engrg.
32 (1982), no. 13, 199–259. FENOMECH
’81, Part I (Stuttgart, 1981). MR 679322
(83k:76005), http://dx.doi.org/10.1016/00457825(82)900718
 [12]
T.
J. R. Hughes, L.
P. Franca, and M.
Mallet, A new finite element formulation for computational fluid
dynamics. I. Symmetric forms of the compressible Euler and NavierStokes
equations and the second law of thermodynamics, Comput. Methods Appl.
Mech. Engrg. 54 (1986), no. 2, 223–234. MR 831553
(87f:76010a), http://dx.doi.org/10.1016/00457825(86)901271
 [13]
C.
Johnson and J.
Pitkäranta, An analysis of the discontinuous
Galerkin method for a scalar hyperbolic equation, Math. Comp. 46 (1986), no. 173, 1–26. MR 815828
(88b:65109), http://dx.doi.org/10.1090/S00255718198608158284
 [14]
Claes
Johnson and Jukka
Saranen, Streamline diffusion methods for the
incompressible Euler and NavierStokes equations, Math. Comp. 47 (1986), no. 175, 1–18. MR 842120
(88b:65133), http://dx.doi.org/10.1090/S00255718198608421204
 [15]
Claes
Johnson and Anders
Szepessy, On the convergence of a finite element
method for a nonlinear hyperbolic conservation law, Math. Comp. 49 (1987), no. 180, 427–444. MR 906180
(88h:65164), http://dx.doi.org/10.1090/S00255718198709061805
 [16]
Claes
Johnson, Anders
Szepessy, and Peter
Hansbo, On the convergence of shockcapturing
streamline diffusion finite element methods for hyperbolic conservation
laws, Math. Comp. 54
(1990), no. 189, 107–129. MR 995210
(90j:65118), http://dx.doi.org/10.1090/S00255718199009952100
 [17]
B. Van Leer, "Towards the ultimate conservation difference scheme, II and V," J. Comput. Phys., v. 14 and 32, 1974 and 1979, pp. 361376 and 101136.
 [18]
P.
Lasaint and P.A.
Raviart, On a finite element method for solving the neutron
transport equation, Mathematical aspects of finite elements in partial
differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin,
Madison, Wis., 1974), Math. Res. Center, Univ. of WisconsinMadison,
Academic Press, New York, 1974, pp. 89–123. Publication No. 33.
MR
0658142 (58 #31918)
 [19]
K. W. Morton & P. K. Sweby, "A comparison of flux limited difference methods and characteristic Galerkin methods for shock modelling," J. Comput. Phys., v. 73, 1987, pp. 203230.
 [20]
Stanley
Osher, Convergence of generalized MUSCL schemes, SIAM J.
Numer. Anal. 22 (1985), no. 5, 947–961. MR 799122
(87b:65147), http://dx.doi.org/10.1137/0722057
 [21]
Stanley
Osher and Sukumar
Chakravarthy, High resolution schemes and the entropy
condition, SIAM J. Numer. Anal. 21 (1984),
no. 5, 955–984. MR 760626
(86a:65086), http://dx.doi.org/10.1137/0721060
 [22]
Stanley
Osher and Eitan
Tadmor, On the convergence of difference
approximations to scalar conservation laws, Math. Comp. 50 (1988), no. 181, 19–51. MR 917817
(89m:65086), http://dx.doi.org/10.1090/S0025571819880917817X
 [23]
Richard
Sanders, A thirdorder accurate variation
nonexpansive difference scheme for single nonlinear conservation
laws, Math. Comp. 51
(1988), no. 184, 535–558. MR 935073
(89c:65104), http://dx.doi.org/10.1090/S00255718198809350733
 [24]
ChiWang
Shu, TVB uniformly highorder schemes for
conservation laws, Math. Comp.
49 (1987), no. 179, 105–121. MR 890256
(89b:65208), http://dx.doi.org/10.1090/S00255718198708902565
 [25]
ChiWang
Shu, TVB boundary treatment for numerical
solutions of conservation laws, Math. Comp.
49 (1987), no. 179, 123–134. MR 890257
(88h:65179), http://dx.doi.org/10.1090/S00255718198708902577
 [26]
ChiWang
Shu, Totalvariationdiminishing time discretizations, SIAM J.
Sci. Statist. Comput. 9 (1988), no. 6,
1073–1084. MR 963855
(90a:65196), http://dx.doi.org/10.1137/0909073
 [27]
ChiWang
Shu and Stanley
Osher, Efficient implementation of essentially nonoscillatory
shockcapturing schemes, J. Comput. Phys. 77 (1988),
no. 2, 439–471. MR 954915
(89g:65113), http://dx.doi.org/10.1016/00219991(88)901775
 [28]
P.
K. Sweby, High resolution schemes using flux limiters for
hyperbolic conservation laws, SIAM J. Numer. Anal. 21
(1984), no. 5, 995–1011. MR 760628
(85m:65085), http://dx.doi.org/10.1137/0721062
 [1]
 A. Bourgeat & B. Cockburn, The TVDprojection method for solving implicit numerical schemes for scalar conservation laws: A numerical study of a simple case, IMA Preprint Series #311, University of Minnesota, April 1987. To appear in SIAM J. Sci. Statist. Comput., March 1989. MR 982223 (90g:65116)
 [2]
 G. Chavent &, B. Cockburn, The local projection discontinuousGalerkin finite element method for scalar conservation laws, IMA Preprint Series #341, University of Minnesota, September 1987, to appear in .
 [3]
 G. Chavent & G. Salzano, "A finite element method for the ID water flooding problem with gravity," J. Comput. Phys., v. 45, 1982, pp. 307344. MR 666166 (83g:76089)
 [4]
 B. Cockburn, The quasimonotone schemes for scalar conservation laws, I, II, and III, IMA Preprint Series #263, 268, 277, University of Minnesota, September and October 1986.
 [5]
 B. Cockburn & C.W. Shu, The RungeKutta local projection discontinuousGalerkin finite element method for scalar conservation laws, IMA Preprint Series #388, University of Minnesota, 1988.
 [6]
 M. Crandall & A. Majda, "Monotone difference approximations for scalar conservation laws," Math. Comp., v. 34, 1980, pp. 121. MR 551288 (81b:65079)
 [7]
 A. Harten, "On a class of high resolution totalvariationstable finitedifference schemes," SIAM J. Numer. Anal., v. 21, 1984, pp. 123. MR 731210 (85f:65085)
 [8]
 A. Harten, Preliminary results on the extension of ENO schemes to twodimensional problems, Proc. Internat. Conf. on Hyperbolic Problems, SaintEtienne, January 1986. MR 910102 (88k:65085)
 [9]
 A. Harten & S. Osher, "Uniformly highorder accurate nonoscillatory schemes, I," SIAM J. Numer. Anal., v. 24, 1987, pp. 279309. MR 881365 (90a:65198)
 [10]
 A. Harten, B. Engquist, S. Osher & S. Chakravarthy, "Uniformly high order accurate essentially nonoscillatory schemes, III," J. Comput. Phys., v. 71, 1987, pp. 231303. MR 897244 (90a:65199)
 [11]
 T. Hughes & A. Brook, "Streamline upwindPetrovGalerkin formulations for convection dominated flows with particular emphasis on the incompressible NavierStokes equations," Comput. Methods Appl. Mech. Engrg., v. 32, 1982, pp. 199259. MR 679322 (83k:76005)
 [12]
 T. Hughes, L. P. Franca, M. Mallet & A. Misukami, "A new finite element formulation for computational fluid dynamics, I, II, III and IV," Comput. Methods Appl. Mech. Engrg., v. 54, 58, 1986, pp. 223234, 341355; pp. 305328, 329336. MR 831553 (87f:76010a)
 [13]
 C. Johnson & J. Pitkäranta, "An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation," Math. Comp., v. 46, 1986, pp. 126. MR 815828 (88b:65109)
 [14]
 C. Johnson & J. Saranen, "Streamline diffusion methods for problems in fluid mechanics," Math. Comp., v. 47, 1986, pp. 118. MR 842120 (88b:65133)
 [15]
 C. Johnson & A. Szepessy, "On the convergence of a finite element method for a nonlinear hyperbolic conservation law," Math. Comp., v. 49, 1987, pp. 427444. MR 906180 (88h:65164)
 [16]
 C. Johnson, A. Szepessy & P. Hansbo, "On the convergence of shock capturing streamline diffusion finite element methods for hyperbolic conservation laws," preprint. MR 995210 (90j:65118)
 [17]
 B. Van Leer, "Towards the ultimate conservation difference scheme, II and V," J. Comput. Phys., v. 14 and 32, 1974 and 1979, pp. 361376 and 101136.
 [18]
 P. Lesaint & P.A. Raviart, "On a finite element method for solving the neutron transport equation," in Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, ed.), Academic Press, 1974, pp. 89145. MR 0658142 (58:31918)
 [19]
 K. W. Morton & P. K. Sweby, "A comparison of flux limited difference methods and characteristic Galerkin methods for shock modelling," J. Comput. Phys., v. 73, 1987, pp. 203230.
 [20]
 S. Osher, "Convergence of generalized MUSCL schemes," SIAM J. Numer. Anal., v. 22, 1985, pp. 947961. MR 799122 (87b:65147)
 [21]
 S. Osher & S. Chakravarthy, "High resolution schemes and the entropy condition," SIAM J. Numer. Anal., v. 21, 1984, pp. 955984. MR 760626 (86a:65086)
 [22]
 S. Osher & E. Tadmor, "On the convergence of difference approximations to scalar conservation laws," Math. Comp., v. 50, 1988, pp. 1951. MR 917817 (89m:65086)
 [23]
 R. Sanders, "A thirdorder accurate variation nonexpansive difference scheme for single nonlinear conservation laws," Math. Comp., v. 51, 1988, pp. 535558. MR 935073 (89c:65104)
 [24]
 C.W. Shu, "TVB uniformly highorder schemes for conservation laws," Math. Comp., v. 49, 1987, pp. 105121. MR 890256 (89b:65208)
 [25]
 C.W. Shu, "TVB boundary treatment for numerical solutions of conservation laws," Math. Comp., v. 49, 1987, pp. 123134. MR 890257 (88h:65179)
 [26]
 C.W. Shu, "TotalVariationDiminishing time discretizations," SIAM J. Sci. Statist. Comput., v. 9, 1988, pp. 10731084. MR 963855 (90a:65196)
 [27]
 C.W. Shu & S. Osher, "Efficient implementation of essentially nonoscillatory shockcapturing schemes," J. Comput. Phys., v. 77, 1988, pp. 439471. MR 954915 (89g:65113)
 [28]
 P. Sweby, "High resolution schemes using flux limiters for hyperbolic conservation laws," SIAM J. Numer. Anal., v. 21, 1984, pp. 9951011. MR 760628 (85m:65085)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909833114
PII:
S 00255718(1989)09833114
Keywords:
TVD,
TVB,
RungeKutta,
discontinuous finite elements,
conservation law
Article copyright:
© Copyright 1989
American Mathematical Society
