The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case
HTML articles powered by AMS MathViewer
- by Bernardo Cockburn, Suchung Hou and Chi-Wang Shu PDF
- Math. Comp. 54 (1990), 545-581 Request permission
Abstract:
In this paper we study the two-dimensional version of the Runge-Kutta Local Projection Discontinuous Galerkin (RKDG) methods, already defined and analyzed in the one-dimensional case. These schemes are defined on general triangulations. They can easily handle the boundary conditions, verify maximum principles, and are formally uniformly high-ordrr accurate. Preliminary numerical results showing the performance of the schemes on a variety of initial-boundary value problems are shown.References
- C. Bardos, A. Y. le Roux, and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 4 (1979), no. 9, 1017–1034. MR 542510, DOI 10.1080/03605307908820117 J. B. Bell, C. N. Dawson, and G. R. Shubin, An unsplit, higher-order Godunov method for scalar conservation laws in multiple dimensions, J. Comput. Phys. 74 (1988), 1-24. G. Chavent, B. Cockburn, G. Cohen, and J. Jaffré, A discontinuous finite element method for nonlinear hyperbolic equations, Innovative Numerical Methods in Engineering (Proc. 4th Internat. Sympos. (Georgia Institute of Technology, Atlanta, Georgia), Springer-Verlag, 1986, pp. 337-342.
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174 B. Cockburn and C.-W. Shu, The Runge—Kutta local projection ${P^1}$ discontinuous Galerkin finite element method for scalar conservation laws, Institute for Mathematics and its Applications Preprint Series # 388, Univ. of Minnesota (1988); Proceedings of First National Fluid Dynamics Congress, Cincinnati, Ohio, July 24-28, 1988; to appear in ${{\text {M}}^2}{\text {AN}}$.
- Bernardo Cockburn and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp. 52 (1989), no. 186, 411–435. MR 983311, DOI 10.1090/S0025-5718-1989-0983311-4
- Bernardo Cockburn, San Yih Lin, and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems, J. Comput. Phys. 84 (1989), no. 1, 90–113. MR 1015355, DOI 10.1016/0021-9991(89)90183-6
- Bernardo Cockburn, Quasimonotone schemes for scalar conservation laws. I, SIAM J. Numer. Anal. 26 (1989), no. 6, 1325–1341. MR 1025091, DOI 10.1137/0726077
- Bernardo Cockburn, Quasimonotone schemes for scalar conservation laws. II, III, SIAM J. Numer. Anal. 27 (1990), no. 1, 247–258, 259–276. MR 1034933, DOI 10.1137/0727017 —, Quasimonotone schemes for scalar conservation laws, III, SIAM J. Numer. Anal. 27 (1990), 259-276.
- 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
- Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682, DOI 10.1007/978-1-4684-9486-0
- Jonathan B. Goodman and Randall J. LeVeque, On the accuracy of stable schemes for $2$D scalar conservation laws, Math. Comp. 45 (1985), no. 171, 15–21. MR 790641, DOI 10.1090/S0025-5718-1985-0790641-4
- 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 —, Preliminary results on the extension of ENO schemes to two-dimensional problems, in Proceedings of the International Conference on Hyperbolic Problems, Saint-Etienne, January 1986.
- Ami Harten, ENO schemes with subcell resolution, J. Comput. Phys. 83 (1989), no. 1, 148–184. MR 1010163, DOI 10.1016/0021-9991(89)90226-X
- 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. Hughes and M. Mallet, A high-precision finite element method for shock-tube calculations, Finite Elements in Fluids, vol. 6 (R. H. Gallagher, G. F. Carey, J. T. Oden, and O. C. Zienkiewicz, eds.), 1985, pp. 339-353.
- 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
- 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
- Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 305–328. MR 865671, DOI 10.1016/0045-7825(86)90152-0
- Thomas J. R. Hughes and Michel Mallet, A new finite element formulation for computational fluid dynamics. III. The generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), no. 3, 305–328. MR 865671, DOI 10.1016/0045-7825(86)90152-0
- 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
- W. B. Lindquist, The scalar Riemann problem in two spatial dimensions: piecewise smoothness of solutions and its breakdown, SIAM J. Math. Anal. 17 (1986), no. 5, 1178–1197. MR 853523, DOI 10.1137/0517082
- W. B. Lindquist, Construction of solutions for two-dimensional Riemann problems, Comput. Math. Appl. Part A 12 (1986), no. 4-5, 615–630. Hyperbolic partial differential equations, III. MR 841991
- Bradley J. Lucier, A moving mesh numerical method for hyperbolic conservation laws, Math. Comp. 46 (1986), no. 173, 59–69. MR 815831, DOI 10.1090/S0025-5718-1986-0815831-4 —, Regularity through approximation for scalar conservation laws, IMA Preprint # 336 (1987).
- 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
- 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
- Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. II, J. Comput. Phys. 83 (1989), no. 1, 32–78. MR 1010162, DOI 10.1016/0021-9991(89)90222-2
- 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
- Tong Zhang and Yu Xi Zheng, Two-dimensional Riemann problem for a single conservation law, Trans. Amer. Math. Soc. 312 (1989), no. 2, 589–619. MR 930070, DOI 10.1090/S0002-9947-1989-0930070-3
- Tong Chang and Gui Qiang Chen, Some fundamental concepts about system of two spatial dimensional conservation laws, Acta Math. Sci. (English Ed.) 6 (1986), no. 4, 463–474. MR 924036, DOI 10.1016/S0252-9602(18)30506-X
- David H. Wagner, The Riemann problem in two space dimensions for a single conservation law, SIAM J. Math. Anal. 14 (1983), no. 3, 534–559. MR 697528, DOI 10.1137/0514045
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Math. Comp. 54 (1990), 545-581
- MSC: Primary 65M60; Secondary 35L65, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-1990-1010597-0
- MathSciNet review: 1010597