Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction

SIAM Journal on Numerical Analysis

Volumn 45, Issue 6, 2007, Pages 2442-2467

  Liu, Yingjie ^a   Shu, Chi Wang ^b   Tadmor, Eitan ^c   Zhang, Mengping ^d  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^b BROWN UNIVERSITY   (United States)

^c Bldg 142   (United States)

^d UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA   (China)

Author keywords

Central scheme; Discontinuous Galerkin method; ENO scheme; MUSCL scheme; TVD scheme

Indexed keywords

CENTRAL SCHEME; DISCONTINUOUS GALERKIN METHOD; ENO SCHEME; MUSCL SCHEME; TVD SCHEME;

CELLS; CYTOLOGY; INCOMPRESSIBLE FLOW; MOTION ESTIMATION; NUMERICAL ANALYSIS; REPAIR; RESTORATION;

GALERKIN METHODS;

EID: 46749092411     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/060666974     Document Type: Article

References (50)

1. R. ABGRALL, On essentially non-oscillatory schemes on unstructured meshes: Analysis and implementation, J. Comput. Phys., 114 (1994), pp. 45-58.
2. P. ARMINJON AND A. ST-CYR, Nessyahu-Tadmor-type central finite volume methods without predictor for 3D Cartesian and unstructured tetrahedral grids, Appl. Numer. Math., 46 (2003), pp. 135-155.
3. P. ARMINJON, M. C. VIALLON, A. MADRANE, AND L. KADDOURI, Discontinuous finite elements and 2-dimensional finite volume versions of the Lax-Friedrichs and Nessyahu-Tadmor difference schemes for compressible flows on unstructured grids, in CFD Review, M. Hafez and K. Oshima, eds., John Wiley, New York, 1997, pp. 241-261.
4. U. ASCHER, S. RUUTH, AND R. J. SPITERI, Implicit-explicit Runge-Kutta methods for time dependent partial differential equations, Appl. Numer. Math., 25 (1997), pp. 151-167.
5. J. BALBAS AND E. TADMOR, CentPack: A Package for High-Resolution Central Schemes for Nonlinear Conservation Laws and Related Problems, http://www.cscamm.umd.edu/centpack.
6. F. BIANCO, G. PUPPO, AND G. RUSSO, High-order central schemes for hyperbolic systems of conservation laws, SIAM J. Sci. Comput., 21 (1999), pp. 294-322.
7. R. BISWAS, K. DEVINE, AND J. FLAHERTY, Parallel, adaptive finite element methods for conservation laws, Appl. Numer. Math., 14 (1994), pp. 255-283.
8. J. BORIS AND D. BOOK, Flux corrected transport. I. SHASTA, a fluid transport algorithm that works, J. Comput. Phys., 11 (1973), pp. 38-69.
9. N. CHEVAUGEON, J. XIN, P. HU, X. LI, D. CLER, J. FLAHERTY, AND M. SHEPHARD, Discontinuous Galerkin methods applied to shock and blast problems, J. Sci. Comput., 22/23 (2005), pp. 227-243.
10. B. COCKBURN, An introduction to the discontinuous Galerkin method for convection-dominated problems, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (Cetraro, 1997), Lecture Notes in Math. 1697, Springer, Berlin, 1998, pp. 151-268.
11. B. COCKBURN, S.-C. HOU, AND C.-W. SHU, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws, Math. Comp., 54 (1990), pp. 545-581.
12. 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.
13. 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), pp. 411-435.
14. 1-discontinuous-Galerkin finite element method for scalar conservation laws, RAIRO Modél. Math. Anal. Numér., 25 (1991), pp. 337-361.
15. B. COCKBURN AND C.-W. SHU, The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, J. Comput. Phys., 141 (1998), pp. 199-224.
16. P. COLELLA AND P. WOODWARD, The piecewise parabolic method (PPM) for gas-dynamical simulation, J. Comput. Phys., 54 (1984), pp. 174-201.
17. J. GLIMM, X. LI, Y. LIU, Z. XU, AND N. ZHAO, Conservative front tracking with improved accuracy, SIAM J. Numer. Anal., 41 (2003), pp. 1926-1947.
18. S. GOTTLIEB, C.-W. SHU, AND E. TADMOR, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), pp. 89-112.
19. A. HARTEN, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), pp. 357-393.
20. A. HARTEN, B. ENGQUIST, S. OSHER, AND S. R. CHAKRAVARTHY, Uniformly high order accuracy essentially non-oscillatory schemes III, J. Comput. Phys., 71 (1987), pp. 231-303.
21. G.-S. JIANG, D. LEVY, C.-T. LIN, S. OSHER, AND E. TADMOR, High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM J. Numer. Anal., 35 (1998), pp. 2147-2168.
22. G.-S. JIANG AND E. TADMOR, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput., 19 (1998), pp. 1892-1917.
23. S. JIN AND Z. XIN, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), pp. 235-276.
24. C. A. KENNEDY AND M. H. CARPENTER, Additive Runge-Kutta schemes for convectiondiffusion-reaction equations, Appl. Numer. Math., 44 (2003), pp. 139-181.
25. A. KURGANOV AND D. LEVY, A third-order semidiscrete central scheme for conservation laws and convection-diffusion equations, SIAM J. Sci. Comput., 22 (2000), pp. 1461-1488.
26. A. KURGANOV AND C.-T. LIN, On the reduction of numerical dissipation in central-upwind schemes, Commun. Comput. Phys., 2 (2007), pp. 141-163.
27. A. KURGANOV, S. NOELLE, AND G. PETROVA, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23 (2001), pp. 707-740.
28. A. KURGANOV AND E. TADMOR, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), pp. 241-282.
29. P. D. LAX AND X.-D. LIU, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19 (1998), pp. 319-340.
30. V. I. LEBEDEV, Explicit difference schemes for solving stiff systems of ODEs and PDEs with complex spectrum, Russian J. Numer. Anal. Math. Modelling, 13 (1998), pp. 107-116.
31. 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. 89-123.
32. D. LEVY, G. PUPPO, AND G. RUSSO, A fourth-order central WENO scheme for multidimensional hyperbolic systems of conservation laws, SIAM J. Sci. Comput., 24 (2002), pp. 480-506.
33. S. F. LIOTTA, V. ROMANO, AND G. RUSSO, Central schemes for balance laws of relaxation type, SIAM J. Numer. Anal., 38 (2000), pp. 1337-1356.
34. X. D. LIU AND S. OSHER, Convex ENO high order multi-dimensional schemes without field by field decomposition or staggered grids, J. Comput. Phys., 142 (1998), pp. 304-330.
35. X. D. LIU, S. OSHER, AND T. CHAN, Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), pp. 408-463.
36. X.-D. LIU AND E. TADMOR, Third order nonoscillatory central scheme for hyperbolic conservation laws, Numer. Math., 79 (1998), pp. 397-425.
37. Y. LIU, Central schemes and central discontinuous Galerkin methods on overlapping cells, Conference on Analysis, Modeling and Computation of PDE and Multiphase Flow, Stony Brook, NY, 2004.
38. Y. LIU, Central schemes on overlapping cells, J. Comput. Phys., 209 (2005), pp. 82-104.
39. A. A. MEDOVIKOV, High order explicit methods for parabolic equations, BIT, 38 (1998), pp. 372-390.
40. H. NESSYAHU AND E. TADMOR, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), pp. 408-463.
41. J. QIU AND C.-W. SHU, Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method. II. Two dimensional case, Comput. & Fluids, 34 (2005), pp. 642-663.
42. J. QIU AND C.-W. SHU, Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM J. Sci. Comput., 26 (2005), pp. 907-929.
43. R. SANDERS AND A. WEISER, High resolution staggered mesh approach for nonlinear hyperbolic systems of conservation laws, J. Comput. Phys., 101 (1992), pp. 314-329.
44. C.-W. SHU, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, A. Quarteroni, ed., Lecture Notes in Math. 1697, Springer, Berlin, 1998, pp. 325-432.
45. C.-W. SHU AND S. OSHER, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), pp. 439-471.
46. C.-W. SHU AND S. OSHER, Efficient implementation of essentially nonoscillatory shock-capturing schemes, II, J. Comput. Phys., 83 (1989), pp. 32-78.
47. H. TANG AND T. TANG, Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal., 41 (2003), pp. 487-515.
48. B. VAN LEER, Towards the ultimate conservative difference scheme I, in Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics, Vol. I, Lecture Notes in Phys. 18, Springer-Verlag, Berlin, 1973, pp. 163-168.
49. B. VAN LEER, Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme, J. Comput. Phys., 14 (1974), pp. 361-370.
50. P. WOODWARD AND P. COLELLA, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), pp. 115-173.