Geodesic
Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method
A. Klöckner1 ; T. Warburton2 ; J. S. Hesthaven3
1 Courant Institute of Mathematical Sciences, New York University, New York, NY 10012
2 Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005
3 Division of Applied Mathematics, Brown University, Providence, RI 02912
Mathematical modelling of natural phenomena, Tome 6 (2011) no. 3, pp. 57-83.

Voir la notice de l'article provenant de la source EDP Sciences

We present a novel, cell-local shock detector for use with discontinuous Galerkin (DG) methods. The output of this detector is a reliably scaled, element-wise smoothness estimate which is suited as a control input to a shock capture mechanism. Using an artificial viscosity in the latter role, we obtain a DG scheme for the numerical solution of nonlinear systems of conservation laws. Building on work by Persson and Peraire, we thoroughly justify the detector’s design and analyze its performance on a number of benchmark problems. We further explain the scaling and smoothing steps necessary to turn the output of the detector into a local, artificial viscosity. We close by providing an extensive array of numerical tests of the detector in use.
DOI : 10.1051/mmnp/20116303

A. Klöckner 1 ; T. Warburton 2 ; J. S. Hesthaven 3

1 Courant Institute of Mathematical Sciences, New York University, New York, NY 10012
2 Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005
3 Division of Applied Mathematics, Brown University, Providence, RI 02912 
@article{MMNP_2011_6_3_a3,
     author = {A. Kl\"ockner and T. Warburton and J. S. Hesthaven},
     title = {Viscous {Shock} {Capturing} in a {Time-Explicit} {Discontinuous} {Galerkin} {Method}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {57--83},
     publisher = {mathdoc},
     volume = {6},
     number = {3},
     year = {2011},
     doi = {10.1051/mmnp/20116303},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20116303/}
}
TY  - JOUR
AU  - A. Klöckner
AU  - T. Warburton
AU  - J. S. Hesthaven
TI  - Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method
JO  - Mathematical modelling of natural phenomena
PY  - 2011
SP  - 57
EP  - 83
VL  - 6
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20116303/
DO  - 10.1051/mmnp/20116303
LA  - en
ID  - MMNP_2011_6_3_a3
ER  - 
%0 Journal Article
%A A. Klöckner
%A T. Warburton
%A J. S. Hesthaven
%T Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method
%J Mathematical modelling of natural phenomena
%D 2011
%P 57-83
%V 6
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20116303/
%R 10.1051/mmnp/20116303
%G en
%F MMNP_2011_6_3_a3
A. Klöckner; T. Warburton; J. S. Hesthaven. Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method. Mathematical modelling of natural phenomena, Tome 6 (2011) no. 3, pp. 57-83. doi : 10.1051/mmnp/20116303. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20116303/

[1] D. N. Arnold, F. Brezzi, B. Cockburn, L. D. Marini SIAM Journal on Numerical Analysis 2002 1749 1779

[2] M. Arora, P. L. Roe Journal of Computational Physics 1997 25 40

[3] ASC Flash Center. Flash user’s guide, version 3.2, Tech. report, University of Chicago, 2009.

[4] G. E. Barter, D. L. Darmofal Journal of Computational Physics 2010 1810 1827

[5] F. Bassi and S. Rebay. Accurate 2D Euler computations by means of a high order discontinuous finite element method. XIVth ICN MFD (Bangalore, India), Springer, 1994.

[6] F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti, M. Savini. A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics (Antwerpen, Belgium) (R. Decuypere and G. Dibelius, eds.), Technologisch Instituut, (1997), 99-108.

[7] N. Bell, M. Garland. Efficient sparse matrix-vector multiplication on CUDA. NVIDIA Technical Report NVR-2008-004, NVIDIA Corporation, 2008.

[8] P. Bogacki, L. F. Shampine Applied Mathematics Letters 1989 321 325

[9] P. Borwein, T. Erdelyi. Polynomials and polynomial inequalities. first ed., Springer, 1995.

[10] A. Burbeau, P. Sagaut, Ch. H. Bruneau Journal of Computational Physics 2001 111 150

[11] E. Burman BIT Numerical Mathematics 2007 715 733

[12] B. Cockburn, J. Guzmán SIAM Journal on Numerical Analysis 2008 1364 1398

[13] B. Cockburn, C. W. Shu Mathematics of Computation 1989 411 435

[14] B. Cockburn, S. Hou, C.-W. Shu Mathematics of Computation 1990 545 581

[15] B. Cockburn, S.-Y. Lin, C.-W. Shu Journal of Computational Physics 1989 90 113

[16] B. Cockburn, C.-W. Shu Journal of Computational Physics 1998 199 224

[17] P. J. Davis. Interpolation and approximation. Blaisdell Pub. Co., 1963.

[18] V. Dolejsi, M. Feistauer, C. Schwab Mathematics and Computers in Simulation 2003 333 346

[19] J. R. Dormand, P. J. Prince Journal of Computational and Applied Mathematics 1980 19 26

[20] M. Dubiner Journal of Scientific Computing 1991 345 390

[21] G. Efraimsson, G. Kreiss SIAM Journal on Numerical Analysis 1999 853 863

[22] A. Ern, A. F. Stephansen, P. Zunino IMA Journal of Numerical Analysis 2009 235

[23] M. Feistauer, V. Kučera Journal of Computational Physics 2007 208 221

[24] J. E. Flaherty, R. M. Loy, M. S. Shephard, B. K. Szymanski, J. D. Teresco, L. H. Ziantz Journal of Parallel and Distributed Computing 1997 139 152

[25] D. Gottlieb, C.-W. Shu SIAM Review 1997 644 668

[26] S. Gottlieb, D. Ketcheson, C.-W. Shu. Strong stability preserving time discretizations. World Scientific, 2011.

[27] P. M. Gresho, R. L. Lee Computers Fluids 1981 223 253

[28] J.-L. Guermond, R. Pasquetti Comptes Rendus Mathematique 2008 801 806

[29] M. Harris. Optimizing parallel reduction in CUDA. Tech. report, Nvidia Corporation, Santa Clara, CA, 2007.

[30] R. Hartmann International Journal for Numerical Methods in Fluids 2006 1131 1156

[31] J. S. Hesthaven, T. Warburton. Nodal discontinuous Galerkin methods: algorithms, analysis, and applications. Springer, 2007.

[32] P. Houston, E. Suli Computer Methods in Applied Mechanics and Engineering 2005 229 243

[33] J. Jaffre, C. Johnson, A. Szepessy Math. Models Methods Appl. Sci. 1995 367 386

[34] V. John, E. Schmeyer Computer Methods in Applied Mechanics and Engineering 2008 475 494

[35] R. M. Kirby, S. J. Sherwin Computer Methods in Applied Mechanics and Engineering 2006 3128 3144

[36] A. Klöckner, T. Warburton, J. Bridge, J. S. Hesthaven J. Comp. Phys. 2009 7863 7882

[37] T. Koornwinder. Two-variable analogues of the classical orthogonal polynomials. Theory and Applications of Special Functions (1975), 435-495.

[38] G. Kreiss, G. Efraimsson, J. Nordstrom SIAM Journal on Numerical Analysis 2001 1986 1998

[39] L. Krivodonova Journal of Computational Physics 2007 879 896

[40] D. Kuzmin, R. Löhner, S. Turek. Flux-corrected transport. Springer, 2005.

[41] A. Lapidus Journal of Computational Physics 1967 154 177

[42] P. D. Lax Communications on Pure and Applied Mathematics 1954 159 193

[43] P. Lesaint, P. A. Raviart. On a finite element method for solving the neutron transport equation. Mathematical aspects of finite elements in partial differential equations, (1974), 89-123.

[44] F. Lorcher, G. Gassner, C.-D. Munz J. Comp. Phys. 2008 5649 5670

[45] C. Mavriplis Computer Methods in Applied Mechanics and Engineering 1994 77 86

[46] P. Persson, J. Peraire. Sub-cell shock capturing for discontinuous Galerkin methods. Proc. of the 44th AIAA Aerospace Sciences Meeting and Exhibit, 112 (2006).

[47] J. Proft, B. Riviere Int. J. Num. Anal. Mod. 2009 533 561

[48] W. H. Reed, T. R. Hill. Triangular mesh methods for the neutron transport equation. Tech. report, Los Alamos Scientific Laboratory, Los Alamos, 1973.

[49] F. Rieper Journal of Computational Physics 2010 221 232

[50] C.-W. Shu, S. Osher Journal of Computational Physics 1989 32 78

[51] C.W. Shu SIAM Journal on Scientific and Statistical Computing 1988 1073 1086

[52] J. W. Slater, J. C. Dudek, K. E. Tatum, et al. The NPARC alliance verification and validation archive. 2009.

[53] G. A. Sod Journal of Computational Physics 1978 1 31

[54] J. M. Stone. Athena test archive. 2009.

[55] E. Tadmor SIAM Journal on Numerical Analysis 1989 30 44

[56] S. Tu, S. Aliabadi International Journal of Numerical Analysis and Modeling 2005 163 178

[57] J. Von Neumann, R. Richtmyer Journal of Applied Physics 1950 232 237

[58] T. Warburton J. Eng. Math. 2006 247 262

[59] T. Warburton, T. Hagstrom SIAM J. Num. Anal. 2008 3151 3180

[60] T. C. Warburton, I. Lomtev, Y. Du, S. J. Sherwin, G. E. Karniadakis Computer Methods in Applied Mechanics and Engineering 1999 343 359

[61] P. Woodward, P. Colella Journal of Computational Physics 1984 115 173

[62] Z. Xu, J. Xu, C.-W. Shu. A high order adaptive finite element method for solving nonlinear hyperbolic conservation laws. Tech. Report 2010-14, Scientific Computing Group, Brown University, Providence, RI, USA, 2010.

[63] Z. Xu, Y. Liu, C.-W. Shu Journal of Computational Physics 2009 2194 2212

[64] Y. C. Zhou, G. W. Wei Journal of Computational Physics 2003 159 179

Cité par Sources :