Viscous stabilization of discontinuous Galerkin solutions of hyperbolic conservation laws

Applied Numerical Mathematics

Volumn 56, Issue 3-4 SPEC. ISS., 2006, Pages 444-458

  Xin, Jianguo ^a   Flaherty, Joseph E ^b  

^a Rensselaer Polytechnic Institute   (United States)

^b RENSSELAER POLYTECHNIC INSTITUTE   (United States)

Author keywords

Discontinuous Galerkin; Hyperbolic conservation laws; Viscous stabilization

Indexed keywords

KINEMATICS; LINEARIZATION; MATHEMATICAL MODELS; OSCILLATIONS; PROBLEM SOLVING; WAVE EQUATIONS;

ARTIFICIAL VISCOSITY; DISCONTINUOUS GALERKIN; HYPERBOLIC CONSERVATION LAWS; VISCOUS STABILIZATION;

VISCOSITY;

EID: 33644629091     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2005.08.001     Document Type: Article

References (39)

1. H. Atkins Steps toward a robust high-order simulation tool for aerospace applications B. Cockburn G. Karniadakis C.-W. Shu Discontinuous Galerkin Methods: Theory, Computation and Applications 2000 Springer Berlin 53 61
2. 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 267 279
3. Z. Chen, B. Cockburn, J. Jerome, and C.-W. Shu Mixed-RKDG finite element methods for the 2-D hydrodynamics model for semiconductor device simulation VLSI Design 3 1995 145 158
4. B. Cockburn, and P.-A. Gremaud Error estimates for finite element methods for scalar conservation laws SIAM J. Numer. Anal. 33 1996 522 554
5. B. Cockburn, S. Hou, and C.-W. Shu The Runge-Kutta local projection discontinuous Galerkin finite element method for the conservation laws IV: The multidimensional case Math. Comp. 54 1990 545 581
6. B. Cockburn G. Karniadakis C.-W. Shu Discontinuous Galerkin Methods: Theory, Computation and Applications Lecture Notes in Comput. Sci. Engrg. vol. 11 2000 Springer Berlin
7. B. Cockburn, F. Li, and C.-W. Shu Locally divergence-free discontinuous Galerkin methods for the Maxell equations J. Comput. Phys. 194 2004 588 610
8. B. Cockburn, S. Lin, and C.-W. Shu TVB Runge-Kutta local projection discontinuous Galerkin methods for scalar conservation laws III: One dimensional systems J. Comput. Phys. 84 1989 90 113
9. B. Cockburn, and C.-W. Shu TVB Runge-Kutta local projection discontinuous Galerkin methods for scalar conservation laws II: General framework Math. Comp. 52 1989 411 435
10. 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 199 224
11. Y. Demay, A. Béliveau, and A. Fortin A numerical method for the deformation of two-dimensional drops with surface tension Internat. J. Comput. Fluid Dynamics 10 1998 225 240
12. C. Farhat, I. Harari, and U. Hetmaniuk A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime Comput. Methods Appl. Mech. Engrg. 192 2003 1389 1419
13. J. Flaherty, R. Loy, M. Shephard, and J. Teresco Software for the parallel adaptive solution of conservation laws by discontinuous Galerkin methods B. Cockburn G. Karniadakis S.-W. Shu Discontinous Galerkin Methods Theory, Computation and Applications Lecture Notes in Comput. Sci. Engrg. vol. 11 2000 Springer Berlin 113 124
14. J. Goodman, and R. LeVeque On the accuracy of stable schemes for 2D scalar conservation laws Math. Comp. 45 1985 15 21
15. A. Harten, S. Osher, B. Engquist, and S. Chacravarthy Uniformly high-order accurate nonoscillatory schemes, III J. Comput. Phys. 71 1987 231 303
16. R. Hartmann, and P. Houston Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws SIAM J. Sci. Comput. 24 2002 979 1004
17. R. Hartmann, and P. Houston Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations J. Comput. Phys. 183 2002 508 532
18. H. Holden, K.-A. Lie, and N.H. Risebro An unconditionally stable method for the Euler equations J. Comput. Phys. 150 1999 76 96
19. P. Houston, I. Perugia, and D. Schötzau Mixed discontinuous Galerkin approximation of the Maxwell operator SIAM J. Numer. Anal. 42 2004 434 459
20. P. Houston, I. Perugia, and D. Schötzau Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator Comput. Methods Appl. Mech. Engrg. 194 2005 499 510
21. J. Jaffre, C. Johnson, and A. Szepessy Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws Math. Models Methods Appl. Sci. 5 1995 367 386
22. G.-S. Jiang, and C.-W. Shu Efficient implementation of weighted ENO schemes J. Comput. Phys. 126 1996 202 228
23. G. Karniadakis, and S. Sherwin Spectral/hp Element Methods for CFD 1999 Oxford University Press New York
24. T. Koornwinder Two-variable analogues of the classical orthogonal polynomials Theory and Applications of Special Functions 1975 Academic Press New York 435 495
25. L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon, and J.E. Flaherty Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws Appl. Numer. Math. 48 2004 323 338
26. A. Kurganov, and E. Tadmor Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers Numer. Methods Partial Differential Equations 18 2002 584 608
27. 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 319 340
28. R.J. LeVeque Finite Volume Methods for Hyperbolic Problems 2002 Cambridge University Press Cambridge
29. J. Qiu, and C.-W. Shu Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: One-dimensional case J. Comput. Phys. 193 2003 115 135
30. J.-F. Remacle, J.E. Flaherty, and M.S. Shephard An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems SIAM Rev. 45 2003 53 72
31. D. Schötzau, C. Schwab, and A. Toselli Mixed hp-DGFEM for incompressible flows SIAM J. Numer. Anal. 40 2003 2171 2194
32. C.W. Schulz-Rinne, J.P. Collins, and H.M. Glaz Numerical solution of the Riemann problem for two-dimensional gas dynamics SIAM J. Sci. Comput. 14 1993 1394 1414
33. C.-W. Shu, and S. Osher Efficient implementation of essentially non-oscillatory shock-capturing schemes, II J. Comput. Phys. 83 1989 32 78
34. G. Sod A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws J. Comput. Phys. 27 1978 1 31
35. E.F. Toro Riemann Solvers and Numerical Methods for Fluid Dynamics 1997 Springer Berlin
36. T. Warburton, and G. Karniadakis A discontinuous Galerkin method for the viscous MHD equations J. Comput. Phys. 152 1999 1 34
37. L.C. Wellford, and J.T. Oden A theory of discontinuous finite element approximations for the analysis of shock waves in nonlinear elastic materials J. Comput. Phys. 19 1975 179 210
38. P. Woodward, and P. Colella The numerical simulation of two-dimensional fluid flow with strong shocks J. Comput. Phys. 54 1984 115 173
39. T. Zhang, and Y. Zheng Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamic systems SIAM J. Math. Anal. 21 1990 593 630