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Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method

Published online by Cambridge University Press:  16 May 2011

A. Klöckner ,
T. Warburton  and
J. S. Hesthaven
A. Klöckner*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012
T. Warburton
Affiliation:
Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005
J. S. Hesthaven
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912
*
⋆ Corresponding author. E-mail: kloeckner@cims.nyu.edu
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Abstract

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.

Keywords

shock detectionEuler’s equationsdiscontinuous Galerkinexplicit time integrationshock capturingartificial viscosity
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 3: Computational aerodynamics , 2011 , pp. 57 - 83
Copyright
© EDP Sciences, 2011

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References

Références

Arnold, D. N., Brezzi, F., Cockburn, B., Marini, L. D.. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis, 39 (2002), No. 5, 1749-1779. CrossRefGoogle Scholar
Arora, M., Roe, P. L.. On postshock oscillations due to shock capturing schemes in unsteady flows. Journal of Computational Physics, 130 (1997), No. 1, 25-40. CrossRefGoogle Scholar
ASC Flash Center. Flash user’s guide, version 3.2, Tech. report, University of Chicago, 2009.
Barter, G. E. and Darmofal, D. L.. Shock capturing with PDE-based artificial viscosity for DGFEM: Part I. Formulation. Journal of Computational Physics, 229 (2010), No. 5, 1810-1827. CrossRefGoogle Scholar
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.
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.
N. Bell, M. Garland. Efficient sparse matrix-vector multiplication on CUDA. NVIDIA Technical Report NVR-2008-004, NVIDIA Corporation, 2008.
Bogacki, P., Shampine, L. F.. A 3(2) pair of Runge-Kutta formulas. Applied Mathematics Letters, 2 (1989), No. 4, 321-325. CrossRefGoogle Scholar
P. Borwein, T. Erdelyi. Polynomials and polynomial inequalities. first ed., Springer, 1995.
Burbeau, A., Sagaut, P., Bruneau, Ch. H.. A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods. Journal of Computational Physics, 169 (2001), No. 1, 111-150. CrossRefGoogle Scholar
Burman, E.. On nonlinear artificial viscosity, discrete maximum principle and hyperbolic conservation laws. BIT Numerical Mathematics, 47 (2007), No. 4, 715-733. CrossRefGoogle Scholar
Cockburn, B., Guzmán, J.. Error estimates for the Runge-Kutta discontinuous Galerkin method for the transport equation with discontinuous initial data. SIAM Journal on Numerical Analysis, 46 (2008), No. 3, 1364-1398. CrossRefGoogle Scholar
Cockburn, B. and Shu, C. W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Mathematics of Computation, 52 (1989), No. 186, 411-435. Google Scholar
Cockburn, B., Hou, S., Shu, C.-W.. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV: the multidimensional case. Mathematics of Computation, 54 (1990), No. 190, 545-581. Google Scholar
Cockburn, B., Lin, S.-Y., Shu, C.-W.. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. Journal of Computational Physics, 84 (1989), No. 1, 90-113. CrossRefGoogle Scholar
Cockburn, B., Shu, C.-W.. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. Journal of Computational Physics, 141 (1998), No. 2, 199-224. CrossRefGoogle Scholar
P. J. Davis. Interpolation and approximation. Blaisdell Pub. Co., 1963.
Dolejsi, V., Feistauer, M., Schwab, C.. On some aspects of the discontinuous Galerkin finite element method for conservation laws. Mathematics and Computers in Simulation, 61 (2003), No. 3-6, 333-346. CrossRefGoogle Scholar
Dormand, J. R., Prince, P. J.. A family of embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics, 6 (1980), No. 1, 19-26. CrossRefGoogle Scholar
Dubiner, M.. Spectral methods on triangles and other domains. Journal of Scientific Computing, 6 (1991), 345-390. CrossRefGoogle Scholar
Efraimsson, G., Kreiss, G.. A remark on numerical errors downstream of slightly viscous shocks. SIAM Journal on Numerical Analysis, 36 (1999), No. 3, 853-863. CrossRefGoogle Scholar
Ern, A., Stephansen, A. F., Zunino, P.. A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity. IMA Journal of Numerical Analysis, 29 (2009), No. 2, 235. CrossRefGoogle Scholar
Feistauer, M., Kučera, V.. On a robust discontinuous Galerkin technique for the solution of compressible flow. Journal of Computational Physics, 224 (2007), No. 1, 208-221. CrossRefGoogle Scholar
Flaherty, J. E., Loy, R. M., Shephard, M. S., Szymanski, B. K., Teresco, J. D., Ziantz, L. H.. Adaptive local refinement with octree load balancing for the parallel solution of three-dimensional conservation laws. Journal of Parallel and Distributed Computing, 47 (1997), No. 2, 139-152. CrossRefGoogle Scholar
Gottlieb, D., Shu, C.-W.. On the Gibbs phenomenon and its resolution. SIAM Review, 39 (1997), No. 4, 644-668. CrossRefGoogle Scholar
S. Gottlieb, D. Ketcheson, C.-W. Shu. Strong stability preserving time discretizations. World Scientific, 2011.
Gresho, P. M., Lee, R. L.. Don’t suppress the wiggles-they’re telling you something!. Computers Fluids, 9 (1981), No. 2, 223-253. CrossRefGoogle Scholar
Guermond, J.-L., Pasquetti, R.. Entropy-based nonlinear viscosity for Fourier approximations of conservation laws. Comptes Rendus Mathematique, 346 (2008), No. 13-14, 801-806. CrossRefGoogle Scholar
M. Harris. Optimizing parallel reduction in CUDA. Tech. report, Nvidia Corporation, Santa Clara, CA, 2007.
Hartmann, R.. Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 51 (2006), No. 9, 1131-1156. CrossRefGoogle Scholar
J. S. Hesthaven, T. Warburton. Nodal discontinuous Galerkin methods: algorithms, analysis, and applications. Springer, 2007.
Houston, P., Suli, E.. A note on the design of hp-adaptive finite element methods for elliptic partial differential equations. Computer Methods in Applied Mechanics and Engineering, 194 (2005), No. 2-5, 229-243. CrossRefGoogle Scholar
Jaffre, J., Johnson, C., and Szepessy, A.. Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Math. Models Methods Appl. Sci., 5 (1995), No. 3, 367-386. CrossRefGoogle Scholar
John, V., Schmeyer, E.. Finite element methods for time-dependent convection-diffusion-reaction equations with small diffusion. Computer Methods in Applied Mechanics and Engineering, 198 (2008), No. 3-4, 475-494. CrossRefGoogle Scholar
Kirby, R. M., Sherwin, S. J.. Stabilisation of spectral/hp element methods through spectral vanishing viscosity: Application to fluid mechanics modelling. Computer Methods in Applied Mechanics and Engineering, 195 (2006), No. 23-24, 3128-3144. CrossRefGoogle Scholar
Klöckner, A., Warburton, T., Bridge, J., Hesthaven, J. S.. Nodal discontinuous Galerkin methods on graphics processors. J. Comp. Phys., 228 (2009), 7863-7882. CrossRefGoogle Scholar
T. Koornwinder. Two-variable analogues of the classical orthogonal polynomials. Theory and Applications of Special Functions (1975), 435-495.
Kreiss, G., Efraimsson, G., Nordstrom, J.. Elimination of first order errors in shock calculations. SIAM Journal on Numerical Analysis, 38 (2001), No. 6, 1986-1998. CrossRefGoogle Scholar
Krivodonova, L.. Limiters for high-order discontinuous Galerkin methods. Journal of Computational Physics, 226 (2007), No. 1, 879-896. CrossRefGoogle Scholar
D. Kuzmin, R. Löhner, S. Turek. Flux-corrected transport. Springer, 2005.
Lapidus, A.. A detached shock calculation by second-order finite differences. Journal of Computational Physics, 2 (1967), No. 2, 154-177. CrossRefGoogle Scholar
Lax, P. D.. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Communications on Pure and Applied Mathematics, 7 (1954), No. 1, 159-193. CrossRefGoogle Scholar
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.
Lorcher, F., Gassner, G., Munz, C.-D.. An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations. J. Comp. Phys., 227 (2008), 5649-5670. CrossRefGoogle Scholar
Mavriplis, C.. Adaptive mesh strategies for the spectral element method. Computer Methods in Applied Mechanics and Engineering, 116 (1994), No. 1-4, 77-86. CrossRefGoogle Scholar
P. Persson, J. Peraire. Sub-cell shock capturing for discontinuous Galerkin methods. Proc. of the 44th AIAA Aerospace Sciences Meeting and Exhibit, 112 (2006).
Proft, J., Riviere, B.. Discontinuous Galerkin methods for convection-diffusion equations for varying and vanishing diffusivity. Int. J. Num. Anal. Mod., 6 (2009), No. 4, 533-561. Google Scholar
W. H. Reed, T. R. Hill. Triangular mesh methods for the neutron transport equation. Tech. report, Los Alamos Scientific Laboratory, Los Alamos, 1973.
Rieper, F.. On the dissipation mechanism of upwind-schemes in the low Mach number regime: A comparison between Roe and HLL. Journal of Computational Physics, 229 (2010), No. 2, 221-232. CrossRefGoogle Scholar
Shu, C.-W., Osher, S.. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics, 83 (1989), No. 1, 32-78. CrossRefGoogle Scholar
Shu, C.W.. Total-variation-diminishing time discretizations. SIAM Journal on Scientific and Statistical Computing, 9 (1988), 1073-1086. CrossRefGoogle Scholar
J. W. Slater, J. C. Dudek, K. E. Tatum, et al. The NPARC alliance verification and validation archive. 2009.
Sod, G. A.. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. Journal of Computational Physics, 27 (1978), No. 1, 1-31. CrossRefGoogle Scholar
J. M. Stone. Athena test archive. 2009.
Tadmor, E.. Convergence of spectral methods for nonlinear conservation laws. SIAM Journal on Numerical Analysis, 26 (1989), No. 1, 30-44. CrossRefGoogle Scholar
Tu, S., Aliabadi, S.. A slope limiting procedure in discontinuous Galerkin finite element method for gas dynamics applications. International Journal of Numerical Analysis and Modeling, 2 (2005), No. 2, 163-178. Google Scholar
von Neumann, J., Richtmyer, R.. A method for the numerical calculation of hydrodynamic shocks. Journal of Applied Physics, 21 (1950), 232-237. CrossRefGoogle Scholar
Warburton, T.. An explicit construction of interpolation nodes on the simplex. J. Eng. Math., 56 (2006), 247-262. CrossRefGoogle Scholar
Warburton, T., Hagstrom, T.. Taming the CFL number for discontinuous Galerkin Methods on structured meshes. SIAM J. Num. Anal., 46 (2008), 3151-3180. CrossRefGoogle Scholar
Warburton, T. C., Lomtev, I., Du, Y., Sherwin, S. J., Karniadakis, G. E.. Galerkin and discontinuous Galerkin spectral/hp methods. Computer Methods in Applied Mechanics and Engineering, 175 (1999), No. 3-4, 343-359. CrossRefGoogle Scholar
Woodward, P., Colella, P.. The numerical simulation of two-dimensional fluid flow with strong shocks. Journal of Computational Physics, 54 (1984), No. 1, 115-173. CrossRefGoogle Scholar
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.
Xu, Z., Liu, Y., Shu, C.-W.. Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells. Journal of Computational Physics, 228 (2009), No. 6, 2194-2212. CrossRefGoogle Scholar
Zhou, Y. C., Wei, G. W.. High resolution conjugate filters for the simulation of flows. Journal of Computational Physics, 189 (2003), No. 1, 159-179. CrossRefGoogle Scholar