An entropy stable, hybridizable discontinuous Galerkin method for the compressible Navier-Stokes equations

Williams, D.

doi:10.1090/mcom/3199

An entropy stable, hybridizable discontinuous Galerkin method for the compressible Navier-Stokes equations
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by D. M. Williams PDF

Math. Comp. 87 (2018), 95-121

Abstract:

This article proves that a particular space-time, hybridizable discontinuous Galerkin method is entropy stable for the compressible Navier-Stokes equations. In order to facilitate the proof, ‘entropy variables’ are utilized to rewrite the compressible Navier-Stokes equations in a symmetric form. The resulting form of the equations is discretized with a hybridizable discontinuous finite element approach in space, and a classical discontinuous finite element approach in time. Thereafter, the initial solution is shown to continually bound the solutions at later times.

References

S. R. Allmaras, V. Venkatakrishnan, and F. T. Johnson, Farfield boundary conditions for 2-d airfoils, (2005), AIAA 2005-4711, 17th AIAA CFD Conference, Toronto, Ontario.
A. Balan, M. Woopen, and G. May, Adjoint-based hp-adaptation for a class of high-order hybridized finite element schemes for compressible flows, (2013), AIAA 2013-2938, 21st AIAA CFD Conference, San Diego, California.
Garrett E. Barter and David L. Darmofal, Shock capturing with PDE-based artificial viscosity for DGFEM. I. Formulation, J. Comput. Phys. 229 (2010), no. 5, 1810–1827. MR 2578253, DOI 10.1016/j.jcp.2009.11.010
Timothy J. Barth, Numerical methods for gasdynamic systems on unstructured meshes, An introduction to recent developments in theory and numerics for conservation laws (Freiburg/Littenweiler, 1997) Lect. Notes Comput. Sci. Eng., vol. 5, Springer, Berlin, 1999, pp. 195–285. MR 1731618, DOI 10.1007/978-3-642-58535-7_{5}
F. Bassi, L. Botti, A. Colombo, A. Crivellini, N. Franchina, A. Ghidoni, and S. Rebay, Very high-order accurate discontinuous Galerkin computation of transonic turbulent flows on aeronautical configurations, ADIGMA-A European Initiative on the Development of Adaptive Higher-Order Variational Methods for Aerospace Applications, Springer, 2010, pp. 25–38.
F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti, and M. Savini, A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows, Proceedings of the 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, Technologisch Instituut, Antwerpen, Belgium, 1997, pp. 99–109.
M. Carpenter and T. Fisher, High-order entropy stable formulations for computational fluid dynamics, (2013), AIAA 2013-2868, 21st AIAA CFD Conference, San Diego, California.
Mark H. Carpenter, Travis C. Fisher, Eric J. Nielsen, and Steven H. Frankel, Entropy stable spectral collocation schemes for the Navier-Stokes equations: discontinuous interfaces, SIAM J. Sci. Comput. 36 (2014), no. 5, B835–B867. MR 3268619, DOI 10.1137/130932193
Bernardo Cockburn, Bo Dong, and Johnny Guzmán, A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. Comp. 77 (2008), no. 264, 1887–1916. MR 2429868, DOI 10.1090/S0025-5718-08-02123-6
Bernardo Cockburn, Bo Dong, Johnny Guzmán, Marco Restelli, and Riccardo Sacco, A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems, SIAM J. Sci. Comput. 31 (2009), no. 5, 3827–3846. MR 2556564, DOI 10.1137/080728810
Bernardo Cockburn and Jayadeep Gopalakrishnan, The derivation of hybridizable discontinuous Galerkin methods for Stokes flow, SIAM J. Numer. Anal. 47 (2009), no. 2, 1092–1125. MR 2485446, DOI 10.1137/080726653
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365. MR 2485455, DOI 10.1137/070706616
Bernardo Cockburn, Johnny Guzmán, See-Chew Soon, and Henry K. Stolarski, An analysis of the embedded discontinuous Galerkin method for second-order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 4, 2686–2707. MR 2551142, DOI 10.1137/080726914
Jim Douglas Jr. and Todd Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975) Lecture Notes in Phys., Vol. 58, Springer, Berlin, 1976, pp. 207–216. MR 0440955
Michael Dumbser, Olindo Zanotti, Raphaël Loubère, and Steven Diot, A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws, J. Comput. Phys. 278 (2014), 47–75. MR 3261082, DOI 10.1016/j.jcp.2014.08.009
Herbert Egger and Christian Waluga, $hp$ analysis of a hybrid DG method for Stokes flow, IMA J. Numer. Anal. 33 (2013), no. 2, 687–721. MR 3047948, DOI 10.1093/imanum/drs018
Herbert Egger and Christian Waluga, A hybrid discontinuous Galerkin method for Darcy-Stokes problems, Domain decomposition methods in science and engineering XX, Lect. Notes Comput. Sci. Eng., vol. 91, Springer, Heidelberg, 2013, pp. 663–670. MR 3243045, DOI 10.1007/978-3-642-35275-1_{7}9
K. Fidkowski, An output-based adaptive hybridized discontinuous Galerkin method on deforming domains, (2015), AIAA 2015-2602, 22nd AIAA CFD Conference, Dallas, Texas.
Travis C. Fisher and Mark H. Carpenter, High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains, J. Comput. Phys. 252 (2013), 518–557. MR 3101520, DOI 10.1016/j.jcp.2013.06.014
Ulrik S. Fjordholm, Siddhartha Mishra, and Eitan Tadmor, Entropy stable ENO scheme, Hyperbolic problems—theory, numerics and applications. Volume 1, Ser. Contemp. Appl. Math. CAM, vol. 17, World Sci. Publishing, Singapore, 2012, pp. 12–27. MR 3014553
Ulrik S. Fjordholm, Siddhartha Mishra, and Eitan Tadmor, Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws, SIAM J. Numer. Anal. 50 (2012), no. 2, 544–573. MR 2914275, DOI 10.1137/110836961
S. K. Godunov, An interesting class of quasi-linear systems, Dokl. Akad. Nauk SSSR 139 (1961), 521–523 (Russian). MR 0131653
Amiram Harten, On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys. 49 (1983), no. 1, 151–164. MR 694161, DOI 10.1016/0021-9991(83)90118-3
R. Hartmann, Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids 51 (2006), no. 9-10, 1131–1156. MR 2242382, DOI 10.1002/fld.1134
R. Hartmann, Higher-order and adaptive discontinuous Galerkin methods with shock-capturing applied to transonic turbulent delta wing flow, Internat. J. Numer. Methods Fluids 72 (2013), no. 8, 883–894. MR 3069932, DOI 10.1002/fld.3762
Ralf Hartmann and Paul Houston, Symmetric interior penalty DG methods for the compressible Navier-Stokes equations. I. Method formulation, Int. J. Numer. Anal. Model. 3 (2006), no. 1, 1–20. MR 2208562
Ralf Hartmann and Paul Houston, An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equations, J. Comput. Phys. 227 (2008), no. 22, 9670–9685. MR 2467638, DOI 10.1016/j.jcp.2008.07.015
Andreas Hiltebrand and Siddhartha Mishra, Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws, Numer. Math. 126 (2014), no. 1, 103–151. MR 3149074, DOI 10.1007/s00211-013-0558-0
Songming Hou and Xu-Dong Liu, Solutions of multi-dimensional hyperbolic systems of conservation laws by square entropy condition satisfying discontinuous Galerkin method, J. Sci. Comput. 31 (2007), no. 1-2, 127–151. MR 2304273, DOI 10.1007/s10915-006-9105-9
A. Huerta, E. Casoni, and J. Peraire, A simple shock-capturing technique for high-order discontinuous Galerkin methods, Internat. J. Numer. Methods Fluids 69 (2012), no. 10, 1614–1632. MR 2950946, DOI 10.1002/fld.2654
T. J. R. Hughes, L. P. Franca, and M. Mallet, A new finite element formulation for computational fluid dynamics. I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics, Comput. Methods Appl. Mech. Engrg. 54 (1986), no. 2, 223–234. MR 831553, DOI 10.1016/0045-7825(86)90127-1
Alexander Jaust and Jochen Schütz, A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows, Comput. & Fluids 98 (2014), 177–185. MR 3209965, DOI 10.1016/j.compfluid.2014.01.019
Guang Shan Jiang and Chi-Wang Shu, On a cell entropy inequality for discontinuous Galerkin methods, Math. Comp. 62 (1994), no. 206, 531–538. MR 1223232, DOI 10.1090/S0025-5718-1994-1223232-7
D. S. Kamenetskiy, On the relation of the embedded discontinuous Galerkin method to the stabilized residual-based finite element methods, Appl. Numer. Math. 108 (2016), 271–285. MR 3528316, DOI 10.1016/j.apnum.2016.01.004
A. Klöckner, T. Warburton, and J. S. Hesthaven, Viscous shock capturing in a time-explicit discontinuous Galerkin method, Math. Model. Nat. Phenom. 6 (2011), no. 3, 57–83. MR 2812626, DOI 10.1051/mmnp/20116303
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), no. 3-4, 323–338. Workshop on Innovative Time Integrators for PDEs. MR 2056921, DOI 10.1016/j.apnum.2003.11.002
Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0350216
C. Lehrenfeld, Hybrid discontinuous Galerkin methods for solving incompressible flow problems, Ph.D. thesis, RWTH Aachen University, 2010.
H. Lim, Y. Yu, J. Glimm, X. L. Li, and D. H. Sharp, Chaos, transport and mesh convergence for fluid mixing, Acta Math. Appl. Sin. Engl. Ser. 24 (2008), no. 3, 355–368. MR 2433866, DOI 10.1007/s10255-008-8019-8
M. S. Mock, Systems of conservation laws of mixed type, J. Differential Equations 37 (1980), no. 1, 70–88. MR 583340, DOI 10.1016/0022-0396(80)90089-3
D. Moro, A hybridizable discontinuous Petrov-Galerkin scheme for compressible flows, Master’s thesis, Massachusetts Institute of Technology, 6 2011.
D. Moro, N. C. Nguyen, and J. Peraire, A hybridized discontinuous Petrov-Galerkin scheme for scalar conservation laws, Internat. J. Numer. Methods Engrg. 91 (2012), no. 9, 950–970. MR 2967261, DOI 10.1002/nme.4300
D. Moro, N. C. Nguyen, J. Peraire, and J. Gopalakrishnan, A hybridizable discontinuous Petrov-Galerkin method for compressible flows, (2011), AIAA 2011-197, 49th AIAA Aerospace Sciences Meeting, Orlando, Florida.
N. C. Nguyen, J. Peraire, and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, J. Comput. Phys. 228 (2009), no. 9, 3232–3254. MR 2513831, DOI 10.1016/j.jcp.2009.01.030
N. C. Nguyen, J. Peraire, and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations, J. Comput. Phys. 228 (2009), no. 23, 8841–8855. MR 2558780, DOI 10.1016/j.jcp.2009.08.030
N. C. Nguyen, J. Peraire, and B. Cockburn, A hybridizable discontinuous Galerkin method for Stokes flow, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 9-12, 582–597. MR 2796169, DOI 10.1016/j.cma.2009.10.007
N. C. Nguyen, J. Peraire, and B. Cockburn, Hybridizable discontinuous Galerkin methods, Spectral and High Order Methods for Partial Differential Equations, Springer, 2011, pp. 63–84.
N. C. Nguyen, J. Peraire, and B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations, J. Comput. Phys. 230 (2011), no. 4, 1147–1170. MR 2753354, DOI 10.1016/j.jcp.2010.10.032
J. Peraire, N. C. Nguyen, and B. Cockburn, A hybridizable discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations, (2010), AIAA 2010-363, 48th AIAA Aerospace Sciences Meeting.
N. C. Nguyen, J. Peraire, and B. Cockburn, An embedded discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations, (2011), AIAA 2011-3228, 20th AIAA CFD Conference, Honolulu, Hawaii.
P-O. Persson and J. Peraire, Sub-cell shock capturing for discontinuous Galerkin methods, (2006), AIAA 2006-112, 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada.
Jianxian Qiu and Chi-Wang Shu, Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM J. Sci. Comput. 26 (2005), no. 3, 907–929. MR 2126118, DOI 10.1137/S1064827503425298
Sander Rhebergen and Bernardo Cockburn, A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains, J. Comput. Phys. 231 (2012), no. 11, 4185–4204. MR 2911790, DOI 10.1016/j.jcp.2012.02.011
Sander Rhebergen, Bernardo Cockburn, and Jaap J. W. van der Vegt, A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations, J. Comput. Phys. 233 (2013), 339–358. MR 3000935, DOI 10.1016/j.jcp.2012.08.052
Farzin Shakib, Thomas J. R. Hughes, and Zdeněk Johan, A new finite element formulation for computational fluid dynamics. X. The compressible Euler and Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 89 (1991), no. 1-3, 141–219. Second World Congress on Computational Mechanics, Part I (Stuttgart, 1990). MR 1132984, DOI 10.1016/0045-7825(91)90041-4
Tayfun E. Tezduyar and Masayoshi Senga, Stabilization and shock-capturing parameters in SUPG formulation of compressible flows, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 13-16, 1621–1632. MR 2203985, DOI 10.1016/j.cma.2005.05.032
C. Waluga, Analysis of hybrid discontinuous Galerkin methods for incompressible flow problems, Ph.D. thesis, RWTH Aachen University, 2012.
M. Woopen and G. May, An anisotropic adjoint-based hp-adaptive HDG method for compressible turbulent flow, (2015), AIAA 2015-2042, 53rd AIAA Aerospace Sciences Meeting, Kissimmee, Florida.
M. Woopen, G. May, and J. Schütz, Adjoint-based error estimation and mesh adaptation for hybridized discontinuous Galerkin methods, Internat. J. Numer. Methods Fluids 76 (2014), no. 11, 811–834. MR 3281586, DOI 10.1002/fld.3959