TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework
HTML articles powered by AMS MathViewer
- by Bernardo Cockburn and Chi-Wang Shu PDF
- Math. Comp. 52 (1989), 411-435 Request permission
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 ${u_t} + \sum \nolimits _{i = 1}^d {{{({f_i}(u))}_{{x_i}}} = 0}$. In this paper we present a general framework of the methods, up to any order of formal accuracy, using scalar one-dimensional initial value and initial-boundary 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.References
- A. Bourgeat and Bernardo Cockburn, A total variation diminishing-projection 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, DOI 10.1137/0910018 G. Chavent &, B. Cockburn, The local projection ${P^0}{P^1}$-discontinuous-Galerkin finite element method for scalar conservation laws, IMA Preprint Series #341, University of Minnesota, September 1987, to appear in ${M^2}AN$.
- G. Chavent and G. Salzano, A finite-element method for the $1$-D water flooding problem with gravity, J. Comput. Phys. 45 (1982), no. 3, 307–344. MR 666166, DOI 10.1016/0021-9991(82)90107-3 B. Cockburn, The quasi-monotone schemes for scalar conservation laws, I, II, and III, IMA Preprint Series #263, 268, 277, University of Minnesota, September and October 1986. B. Cockburn & C.-W. Shu, The Runge-Kutta local projection ${P^1}$-discontinuous-Galerkin finite element method for scalar conservation laws, IMA Preprint Series #388, University of Minnesota, 1988.
- Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1–21. MR 551288, DOI 10.1090/S0025-5718-1980-0551288-3
- Ami Harten, On a class of high resolution total-variation-stable finite-difference schemes, SIAM J. Numer. Anal. 21 (1984), no. 1, 1–23. With an appendix by Peter D. Lax. MR 731210, DOI 10.1137/0721001
- Ami Harten, Preliminary results on the extension of ENO schemes to two-dimensional problems, Nonlinear hyperbolic problems (St. Etienne, 1986) Lecture Notes in Math., vol. 1270, Springer, Berlin, 1987, pp. 23–40. MR 910102, DOI 10.1007/BFb0078315
- Ami Harten and Stanley Osher, Uniformly high-order accurate nonoscillatory schemes. I, SIAM J. Numer. Anal. 24 (1987), no. 2, 279–309. MR 881365, DOI 10.1137/0724022
- Ami Harten, Björn Engquist, Stanley Osher, and Sukumar R. Chakravarthy, Uniformly high-order accurate essentially nonoscillatory schemes. III, J. Comput. Phys. 71 (1987), no. 2, 231–303. MR 897244, DOI 10.1016/0021-9991(87)90031-3
- Alexander N. Brooks and Thomas J. R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982), no. 1-3, 199–259. FENOMECH ”81, Part I (Stuttgart, 1981). MR 679322, DOI 10.1016/0045-7825(82)90071-8
- 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 Navier-Stokes equations and the second law of thermodynamics, Comput. Methods Appl. Mech. Engrg. 54 (1986), no. 2, 223–234. MR 831553, DOI 10.1016/0045-7825(86)90127-1
- 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, DOI 10.1090/S0025-5718-1986-0815828-4
- Claes Johnson and Jukka Saranen, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations, Math. Comp. 47 (1986), no. 175, 1–18. MR 842120, DOI 10.1090/S0025-5718-1986-0842120-4
- 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, DOI 10.1090/S0025-5718-1987-0906180-5
- Claes Johnson, Anders Szepessy, and Peter Hansbo, On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp. 54 (1990), no. 189, 107–129. MR 995210, DOI 10.1090/S0025-5718-1990-0995210-0 B. Van Leer, "Towards the ultimate conservation difference scheme, II and V," J. Comput. Phys., v. 14 and 32, 1974 and 1979, pp. 361-376 and 101-136.
- 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) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89–123. MR 0658142 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. 203-230.
- Stanley Osher, Convergence of generalized MUSCL schemes, SIAM J. Numer. Anal. 22 (1985), no. 5, 947–961. MR 799122, DOI 10.1137/0722057
- Stanley Osher and Sukumar Chakravarthy, High resolution schemes and the entropy condition, SIAM J. Numer. Anal. 21 (1984), no. 5, 955–984. MR 760626, DOI 10.1137/0721060
- 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, DOI 10.1090/S0025-5718-1988-0917817-X
- Richard Sanders, A third-order accurate variation nonexpansive difference scheme for single nonlinear conservation laws, Math. Comp. 51 (1988), no. 184, 535–558. MR 935073, DOI 10.1090/S0025-5718-1988-0935073-3
- Chi-Wang Shu, TVB uniformly high-order schemes for conservation laws, Math. Comp. 49 (1987), no. 179, 105–121. MR 890256, DOI 10.1090/S0025-5718-1987-0890256-5
- Chi-Wang Shu, TVB boundary treatment for numerical solutions of conservation laws, Math. Comp. 49 (1987), no. 179, 123–134. MR 890257, DOI 10.1090/S0025-5718-1987-0890257-7
- Chi-Wang Shu, Total-variation-diminishing time discretizations, SIAM J. Sci. Statist. Comput. 9 (1988), no. 6, 1073–1084. MR 963855, DOI 10.1137/0909073
- Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), no. 2, 439–471. MR 954915, DOI 10.1016/0021-9991(88)90177-5
- 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, DOI 10.1137/0721062
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Math. Comp. 52 (1989), 411-435
- MSC: Primary 65M60; Secondary 35L65, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-1989-0983311-4
- MathSciNet review: 983311