The local discontinuous galerkin method for time-dependent convection-diffusion systems

SIAM Journal on Numerical Analysis

Volumn 35, Issue 6, 1998, Pages 2440-2463

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

^a UNIVERSITY OF MINNESOTA   (United States)

^b BROWN UNIVERSITY   (United States)

Author keywords

Convection diffusion problems; Discontinuous finite elements

Indexed keywords

EID: 0344393366     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142997316712     Document Type: Article

References (41)

1. T. ARBOGAST AND M. F. WHEELER, A characteristics-mixed finite element method for advection-dominated transport problems, SIAM J. Numer. Anal., 32 (1995), pp. 404-424.
2. H. L. ATKINS AND C.-W. SHU, Quadrature-free implementation of discontinuous Galerkin methods for hyperbolic equations, AIAA J., 36 (1998), pp. 775-782.
3. G. A. BAKER, W. X. JUREIDINI, AND O. A. KARAKASHIAN, Piecewise solenoidal vector fields and the Stokes problem, SIAM J. Numer. Anal., 27 (1990), pp. 1466-1485.
4. F. BASSI AND S. REBAY, High-order accurate discontinuous finite element solution of the 2D Euler equations, J. Comput. Phys., to appear.
5. F. BASSI AND S. REBAY, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), pp. 267-279.
6. F. BASSI, S. REBAY, M. SAVINI, G. MARIOTTI, AND S. PEDINOTTI, A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows, in Proc. Second European Conference ASME on Turbomachinery Fluid Dynamics and Thermodynamics, 1995.
7. K. S. BEY AND J. T. ODEN, A Runge-Kutta discontinuous Galerkin finite element method for high speed flows, AIAA 10th Computational Fluid Dynamics Conference, Honolulu, HI, June 24-27, 1991.
8. R. BISWAS, K. D. DEVINE. AND J. FLAHERTY, Parallel, adaptive finite element methods for conservation laws, Appl. Numer. Math., 14 (1994), pp. 255-283.
9. Z. CHEN, B. COCKBURN, J. JEROME, AND C.-W. SHU, Mixed-RKDG finite element method for the drift-diffusion semiconductor device equations, VLSI Design, 3 (1995), pp. 145-158.
10. Z. CHEN, B. COCKBURN, C. GARDNER, AND J. JEROME, Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode, J. Comput. Phys., 117 (1995), pp. 274-280.
11. P. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1975.
12. 2AN, 25 (1991), pp. 337-361.
13. B. COCKBURN AND C. W. SHU, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework, Math. Comp., 52 (1989), pp. 411-435.
14. 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), pp. 90-113.
15. B. COCKBURN, S. Hou, AND C. W. SHU, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, Math. Comp., 54 (1990), pp. 545-581.
16. B. COCKBURN AND C. W. SHU, The Runge-Kutta discontinuous Galerkin finite element method for conservation laws V: Multidimensional systems, J. Comput. Phys., 141 (1998), pp. 199-224.
17. C. N. DAWSON, Godunov-mixed methods for advection-diffusion equations in multidimensions, SIAM J. Numer. Anal., 30 (1993), pp. 1315-1332.
18. H. L. DECOUGNY, K. D. DEVINE, J. E. FLAHERTY, R. M. LOY, C. ÖZTURAN, AND M. S. SHEPHARD, Load balancing for the parallel adaptive solution of partial differential equations, Appl. Numer. Math., 16 (1994), pp. 157-182.
19. M. DELFOUR, W. HAGER, AND F. TROCHU, Discontinuous Galerkin methods for ordinary differential equations, Math. Comp., 36 (1981), pp. 455-473.
20. K. D. DEVINE, J. E. FLAHERTY, R. M. LOY, AND S. R. WHEAT, Parallel Partitioning Strategies for the Adaptive Solution of Conservation Laws, Rensselaer Polytechnic Institute report 94-1, Troy, NY, 1994.
21. K. D. DEVINE, J. E. FLAHERTY, S. R. WHEAT, AND A. B. MACCABE, A massively parallel adaptive finite element method with dynamic load balancing, in Proceeding Supercomputing '93, pp. 2-11.
22. K. ERIKSSON AND C. JOHNSON, Adaptive finite element methods for parabolic problems I: A linear model problem, SIAM J. Numer. Anal., 28 (1991), pp. 43-77.
23. 2 and L∞L∞, SIAM J. Numer. Anal., 32 (1995), pp. 706-740.
24. K. ERIKSSON AND C. JOHNSON, Adaptive finite element methods for parabolic problems IV: A nonlinear problem, SIAM J. Numer. Anal., 32 (1995), pp. 1729-1749.
25. K. ERIKSSON AND C. JOHNSON, Adaptive finite element methods for parabolic problems V: Long time integration, SIAM J. Numer. Anal., 32 (1995), pp. 1750-1763.
26. K. ERIKSSON, C. JOHNSON, AND V. THOMËE, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modël. Math. Anal. Numër., 19 (1985), pp. 611-643.
27. P. JAMET, Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain, SIAM J. Numer. Anal., 15 (1978), pp. 912-928.
28. G. JIANG AND C.-W. SHU, On cell entropy inequality for discontinuous Galerkin methods, Math Comp., 62 (1994), pp. 531-538.
29. C. JOHNSON AND J. PITKÄRANTA, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46 (1986), pp. 1-26.
30. P. LESAINT AND 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, New York, 1974, pp. 89-145.
31. R. B. LOWRIE, P. L. ROE, AND B. VANLEER, A space-time discontinuous Galerkin method for the time accurate numerical solution of hyperbolic conservation laws, 12th AIAA Computational Fluid Dynamics Conference, San Diego, CA, June 19-22, 1995.
32. CH. G. MAKRIDAKIS AND I. BABUŠKA, On the stability of the discontinuous Galerkin method for the heat equation, SIAM J. Numer. Anal., 34 (1997), pp. 389-401.
33. J. T. ODEN, I. BABUŠKA, AND C. E. BAUMANN, A discontinuous hp finite element method for diffusion problems, J. Comput. Phys., 146 (1998), pp. 495-519.
34. S. OSHER, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal., 21 (1984), pp. 217-235.
35. C. ÖZTURAN, H. L. DECOUGNY, M. S. SHEPHARD, AND J. E. FLAHERTY, Parallel adaptive mesh refinement and redistribution on distributed memory computers, Comput. Methods Appl. Mech. Engrg., 119 (1994), pp. 123-137.
36. T. PETERSON, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. Numer. Anal., 28 (1991), pp. 133-140.
37. W. H. REED AND T. R. HILL, Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Laboratory report LA-UR-73-479, Los Alamos, NM, 1973.
38. G. R. RICHTER, An optimal-order error estimate for the discontinuous Galerkin method, Math. Comp., 50 (1988), pp. 75-88.
39. C.-W. SHU, Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 1073-1084.
40. C.-W. SHU AND S. OSHER, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), pp. 439-471.
41. C.-W. SHU AND S. OSHER, Efficient implementation of essentially non-oscillatory shock capturing schemes, II, J. Comp. Phys., 83 (1989), pp. 32-78.