RKDG FINITE element method combined with BGK scheme for solving fluid dynamics system

SIAM Journal on Scientific Computing

Volumn 28, Issue 3, 2006, Pages 805-831

  Dai, Qingfang ^a   Yu, Xijun ^a  

^a INSTITUTE OF APPLIED PHYSICS AND COMPUTATIONAL MATHEMATICS   (China)

Author keywords

Bhatnagar Gross Krook method; Discontinuous galerkin method; Euler equations; Runge kutta method

Indexed keywords

EULER EQUATIONS; FINITE ELEMENT METHOD; GALERKIN METHODS; NUMERICAL METHODS; PROBLEM SOLVING; RUNGE KUTTA METHODS;

BHATNAGAR-GROSS-KROOK METHOD; DISCONTINUOUS GALERKIN METHOD;

FLUID DYNAMICS;

EID: 34249082771     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/040618783     Document Type: Article

References (38)

1. A. D. KOTELNIKOV AND D. C. MONTGOMERY, A kinetic method for computing inhomogeneous fluid behavior, J, Comput. Phys., 134 (1997), pp. 364-388.
2. A. D. KOTELNIKOV AND D. C. MONTGOMERY, Shock induced turbulence in composite materials at moderate Reynolds numbers, Phys. Fluids, 10 (1998), p. 2037.
3. A. HARTEN, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), pp. 357-393.
4. A. HARTEN, B. ENGQUIST, S. OSHER, AND S. R. CHAKRAVARTHY, Uniformly high order accurate essentially non-oscillatory schemes III, J. Comput. Phys., 71 (1987), pp. 231-303.
5. 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.
6. B. COCKBURN AND S.-Y. LIN, 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.
7. 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. 541-581.
8. 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.
9. B. COCKBURN AND C.-W. SHU, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), pp. 2440-2463.
10. B. COCKBURN, G. E. KARNIADAKIS, AND C. W. SHU, EDS., Discontinuous Galerkin methods. Theory, computation and applications, Lect. Notes Comput. Sci. Eng. 11, Springer-Verlag, Berlin, 2000.
11. B. COCKBURN, G. KANSCHAT, I. PERUGIA, AND D. SCHÖTZAU, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal., 39 (2001), pp. 264-285.
12. B. PERTHAME, Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions, SIAM J. Numer. Anal., 29 (1992), pp. 1-19.
13. C. A. KIM AND A. JAMESON, A robust and accurate LED-BGK solver on unstructured adaptive meshes, 3. Comput. Phys., 143 (1998), pp. 598-627.
14. D. I. PULLIN, Direct simulation methods for compressible inviscid ideal gas flow, 3. Comput. Phys., 34 (1980), pp. 231-244.
15. G. H. XU, Hydrodynamic and n-body schemes on an unstructured, adaptive mesh with application to cosmological simulations, Monthly Notices Roy. Astronom. Soc. 288 (1997), pp. 903-919.
16. K. H. PRENDERGAST AND K. XU, Numerical hydrodynamics from gas-kinetic theory, 3. Comput. Phys., 109 (1993), pp. 53-66.
17. K. XU, Numerical Hydrodynamics from Gas-Kinetic Theory, Ph.D. thesis, Columbia University, New York, NY, 1993.
18. K. XU AND K. H. PRENDERGAST, Numerical Navier-Stokes solutions from gas kinetic theory, 3. Comput. Phys., 114 (1994), pp. 9-17.
19. K. XU, L. MARTINELLI, AND A. JAMESON, Gas-kinetic finite volume methods, flux-vector splitting and artificial diffusion, J. Comput. Phys., 120 (1995), pp. 48-65.
20. K. XU AND A. JAMESON, Gas-kinetic relaxation (BGK-type) schemes for the compressible Euler equation, in Proceedings of the 12th AIAA CFD Conference, San Diego, CA, 1995, paper 95-1736.
21. K. XU, C. KIM, L. MARTINELLI, AND A. JAMESON, BGK-based schemes for the simulation of compressible flow, Int. J. Comput. Fluid Dyn., 7 (1996), pp. 213-235.
22. K. XU, BGK-based schemes for multicomponent flow calculation, 3. Comput. Phys., 134 (1997), pp. 122-133.
23. K. XU, Gas-Kinetic Schemes for Unsteady Compressible Flow Simulations, KVI report 1998-03, von Karman Institute for Fluid Dynamics, Rhode-Saint-Genese, Belgium, 1998.
24. K. XU, A gas-kinetic scheme for the Euler equations with heat transfer, SIAM J. Sci. Comput., 20 (1999), pp. 1317-1335.
25. M. SU, K. XU, AND M. S. GHIDAOUI, Low speed flow simulation by the gas-kinetic scheme, J. Comput. Phys., 150 (1998), pp. 17-39.
26. M. N. MACROSSAN, The equilibrium flux method for the calculation of flows with non-equilibrium chemical reactions, 3. Comput. Phys., 80 (1989), pp. 204-231.
27. P. R. WOODWARD AND P. COLELLA, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), pp. 115-173.
28. Q. DAI AND X. YU, RKDG finite element schemes combined with gas-kinetic methods for 1D Euler compressible equations, Math. Numer. Sin., 25 (2003), pp. 407-422.
29. R. D. REITZ, One-dimensional compressible gas dynamics calculations using the Boltzmann equations, 3. Comput. Phys., 42 (1981), pp. 108-123.
30. R. H. SANDERS AND K. H. PRENDERGAST, The possible relation of the three-kiloparsec arm to explosions in the galactic nucleus, Astrophys. J., 188 (1974), pp. 489-500.
31. S. M. DESHPANDE, A Second Order Accurate, Kinetic-Theory Based, Method for Inviscoid Compressible Flows, NASA Langley Tech. paper 2613, NASA Langley Research Center, Hampton, VA, 1986.
32. W. EPPARD AND B. GROSSMAN, A multidimensional kinetic-based upwind solver for the Euler equations, in Proceedings of the 11th AIAA CFD Conference, Orlando, FL, 1993.
33. W. HUAMO, New accurate high resolution TVD schemes, analysis and applications, in Proceedings of the International Symposium on Computational Fluid Dynamics, Nagoya, Japan, 1989.
34. 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 Scientific Laboratory, Los Alamos, NM, 1973.
35. X. YU AND Q. DAI, RKDG finite element method combined with KFVS scheme for solving fluid dynamics system, Numer. Methods Partial Differential Equations, to appear.
36. X. YU AND Q. DAI, RKDG finite element schemes combined with a gas-kinetic method for one-dimensional compressible Euler equations, in Current Trends in Scientific Computing, Contemp. Math. 329, AMS, Providence, RI, 2003, pp. 355-372.
37. X. YU AND Y. WU, A Taylor-Galerkin finite element method for one-dimensional hyperbolic conservation laws, in Fluid Flow and Transport in Porous Media: Mathematical and Numerical Treatment, Contemp. Math. 295, AMS, Providence, RI, pp. 501-518.
38. H. X. ZHANG AND F. G. ZHUANG, NND schemes and their applications to numerical simulation of two- and three-dimensional flows, Adv. in Appl. Mech., 29 (1992), pp. 193-256.