The runge-kuttal ocal projection discontinuous galerkin finite element method for conservation laws iv: The multidimensional case

Mathematics of Computation

Volumn 54, Issue 190, 1990, Pages 545-581

  Cockburn, Bernardo ^a,b   Hou, Suchung ^a,b   Shu, Chi Wang ^a,b  

^a UNIVERSITY OF MINNESOTA   (United States)

^b BROWN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84966261380     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-1990-1010597-0     Document Type: Article

References (43)

1. C. Bardos, A. Y. LeRoux and J. C. Nedelec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 4 (1979), 1017-1034.
2. 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.
3. G. Chavent, B. Cockburn, G. Cohen, and J. Jaffre, A discontinuous finite element method for nonlinear hyperbolic equations, Innovative Numerical Methods in Engineering (Proc. 4th Intemat. Sympos. Georgia Institute of Technology, Atlanta, Georgia), Springer-Verlag, 1986, pp. 337-342.
4. P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1975.
5. B. Cockburn and C.-W. Shu, The Runge-Kutta local projection P' 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 M2AN.
6. B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework. Math. Comp. 52 (1989), 411-435
7. B. Cockburn, S.-Y. Lin, and C.-W. Shu, TVB Runge-Kutta local projection discontinuous-Galerkin finite element method for conservation laws III: One-dimensional systems, J. Comput. Phys. 84 (1989), 90-113.
8. B. Cockburn, Quasimonotone schemes for scalar conservation laws, I, SIAM J. Numer. Anal. 26 (1989), 1325-1341.
9. B. Cockburn, Quasimonotone schemes for scalar conservation laws, II, SIAM J. Numer. Anal. 27 (1990), 247-258.
10. B. Cockburn, Quasimonotone schemes for scalar conservation laws. III, SIAM J. Numer. Anal. 27 (1990), 259-276.
11. M. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), 1-21.
12. E. Giusti, Minimal surfaces and functions of bounded variation, Birkhauser, Basel, 1984.
13. J. Goodman and R. LeVeque, On the accuracy of stable schemes for 2D scalar conservation laws, Math. Comp. 45 (1985), 15-21.
14. A. Harten, On a class ofhigh-resolution total-variation-stablefinite-differenceschemes, SIAM J. Numer. Anal. 21 (1984), 1-23.
15. A. Harten, 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.
16. A. Harten, ENO schemes with subcell resolutions, J. Comput. Phys. 83 (1989), pp. 148-184.
17. A. Harten and S. Osher, Uniformly high-order accurate nonoscillatory schemes, I, SIAM J. Numer. Anal. 24 (1987), 219-309.
18. A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, Uniformly high-order accurate nonoscillatory schemes, III, J. Comput. Phys. 71 (1987), 231-303.
19. T. Hughes and A. Brook, 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), 199-259.
20. 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.
21. T. Hughes, L. P. Franca, M. Mallet and A. Misukami, A new finite element formulation for computational fluid dynamics, I, Comput. Methods Appl. Mech. Engrg. 54 (1986), 223-234.
22. T. Hughes, L. P. Franca, M. Mallet and A. Misukami, A new finite element formulation for computational fluid dynamics, II, Comput. Methods Appl. Mech. Engrg. 54 (1986), 341-355.
23. T. Hughes, L. P. Franca, M. Mallet and A. Misukami, A new finite element formulation for computational fluid dynamics, III, Comput. Methods Appl. Mech. Engrg. 58 (1986), 305-328.
24. T. Hughes, L. P. Franca, M. Mallet and A. Misukami, A new finite element formulation for computational fluid dynamics, IV, Comput. Methods Appl. Mech. Engrg. 58 (1986), 329-336.
25. C. Johnson and J. Saranen, Streamline diffusion methods for problems in fluid mechanics, Math. Comp. 47 (1986), 1-18.
26. C. Johnson and A. Szepessy, On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp. 49 (1987), 427-444.
27. C. Johnson, A. Szepessy and P. Hansbo, On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp. 54 (1990), 107-129.
28. W. B. Lindquist, The scalar Riemann problem in two spatial dimensions: piecewise smoothness of solutions and its breakdown, SIAM J. Math. Anal. 17 (1986), 1178-1197.
29. W. B. Lindquist, Construction of solutions for two-dimensional Riemann problems, Comput. Math. Appl. 12A (1986), 615-630.
30. B. J. Lucier, A moving mesh numerical method for hyperbolic conservation laws. Math. Comp. 46 (1986), 59-69.
31. B. J. Lucier, Regularity through approximation for scalar conservation laws, IMA Preprint # 336 (1987).
32. S. Osher, Convergence of generalized MUSCL schemes, SIAM J. Numer. Anal. 22 (1984), 947-961.
33. S. Osher and S. Chakravarthy, High resolution schemes and the entropy condition, SIAM J. Numer. Anal. 21 (1984), 955-984.
34. R. Sanders, A third-order accurate variation nonexpansive difference scheme for single nonlinear conservation laws, Math. Comp. 51 (1988), 535-558.
35. C.-W. Shu, TVB uniformly high-order schemes for conservation laws, Math, of Comp. 49 (1987), 105-121.
36. C.-W. Shu, TVB boundary treatment for numerical solutions of conservation laws, Math. Comp. 49 (1987), 123-134.
37. C.-W. Shu, TVD time discretizations, SIAM J. Sci. Statist. Comput. 9 (1988), 1073-1084.
38. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), 439-471.
39. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. Comput. Phys. 83 (1989), 32-78.
40. P. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21 (1984), 995-1011
41. C. Tong and Z. Yu-Xi, Two dimensional Riemann problems for a single conservation law. Institute of Mathematics, Academia Sinica, Preprint # 20 (1985).
42. C. Tong and C. Guiquiang, Some fundamental concepts about systems of two spatial dimensional conservation laws, Acta Math. Sci. 6 (1986), 463-474.
43. D. Wagner, The Riemann problem in two space dimensions for a single conservation law, SIAM J. Math. Anal. 14 (1983), 534-559.