A class of embedded discontinuous Galerkin methods for computational fluid dynamics. (English) Zbl 1349.76245
Summary: We present a class of embedded discontinuous Galerkin (EDG) methods for numerically solving the Euler equations and the Navier-Stokes equations. The essential ingredients are a local Galerkin projection of the underlying governing equations at the element level onto spaces of polynomials of degree \(k\) to parametrize the numerical solution in terms of the approximate trace, a judicious choice of the numerical flux to provide stability and consistency, and a global jump condition that weakly enforces the single-valuedness of the numerical flux to arrive at a global formulation in terms of the numerical trace. The EDG methods are thus obtained from the hybridizable discontinuous Galerkin (HDG) method by requiring the approximate trace to belong to smaller approximation spaces than the one in the HDG method. In the EDG methods, the numerical trace is taken to be continuous on a suitable collection of faces, thus resulting in an even smaller number of globally coupled degrees of freedom than in the HDG method. On the other hand, the EDG methods are no longer locally conservative. In the framework of convection-diffusion problems, this lack of local conservativity is reflected in the fact that the EDG methods do not provide the optimal convergence of the approximate gradient or the superconvergence for the scalar variable for diffusion-dominated problems as the HDG method does. However, since the HDG method does not display these properties in the convection-dominated regime, the EDG method becomes a reasonable alternative since it produces smaller algebraic systems than the HDG method. In fact, the resulting stiffness matrix has a similar sparsity pattern as that of the statically condensed continuous Galerkin (CG) method. The main advantage of the EDG methods is that they are generally more stable and robust than the CG method for solving convection-dominated problems. Numerical results are presented to illustrate the performance of the EDG methods. They confirm that, even though the EDG methods are not locally conservative, they are a viable alternative to the HDG method in the convection-dominated regime.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 76N15 | Gas dynamics (general theory) |
Keywords:
finite element methods; discontinuous Galerkin methods; hybrid/mixed methods; Euler equations; Navier-Stokes equations; computational fluid dynamicsReferences:
| [1] | Alexander, R., Diagonally implicit Runge-Kutta methods for stiff ODEs, SIAM J. Numer. Anal., 14, 1006-1021 (1977) · Zbl 0374.65038 |
| [2] | Anderson, D. A.; Tannehill, J. C.; Pletcher, R. H., Computational Fluid Dynamics and Heat Transfer (1984), Hemisphere Publishing: Hemisphere Publishing New York · Zbl 0569.76001 |
| [3] | Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 5, 1749-1779 (2001) · Zbl 1008.65080 |
| [4] | Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131, 2, 267-279 (1997) · Zbl 0871.76040 |
| [5] | Baumann, C.; Oden, J., A discontinuous hp finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Eng., 175, 311-341 (1999) · Zbl 0924.76051 |
| [6] | Brezzi, F.; Douglas, J.; Marini, L. D., Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47, 217-235 (1985) · Zbl 0599.65072 |
| [7] | Buffa, A.; Hughes, T. J.R.; Sangalli, G., Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems, SIAM J. Numer. Anal., 44, 1420-1440 (2006) · Zbl 1153.76038 |
| [8] | Chen, Y.; Cockburn, B., Analysis of variable-degree HDG methods for convection-diffusion equations. Part I: General nonconforming meshes, IMA J. Numer. Anal., 32, 1267-1293 (2012) · Zbl 1277.65094 |
| [9] | Chen, Y.; Cockburn, B., Analysis of variable-degree HDG methods for convection-diffusion equations. Part II: Semimatching nonconforming meshes, Math. Comput., 83, 87-111 (2014) · Zbl 1457.65187 |
| [10] | Chiocchia, G., Exact solutions to transonic and supersonic flows (1985), AGARD, Technical Report AR-211 |
| [11] | Cockburn, B.; Dong, B.; Guzmán, J., A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. Comput., 77, 1887-1916 (2008) · Zbl 1198.65193 |
| [12] | Cockburn, B.; Dong, B.; Guzmán, J.; Restelli, M.; Sacco, R., A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems, SIAM J. Sci. Comput., 31, 5, 3827-3846 (2009) · Zbl 1200.65093 |
| [13] | Cockburn, B.; Gopalakrishnan, J., The derivation of hybridizable discontinuous Galerkin methods for Stokes flow, SIAM J. Numer. Anal., 47, 1092-1125 (2009) · Zbl 1279.76016 |
| [14] | Cockburn, B.; Gopalakrishnan, J.; Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47, 1319-1365 (2009) · Zbl 1205.65312 |
| [15] | Cockburn, B.; Gopalakrishnan, J.; Nguyen, N. C.; Peraire, J.; Sayas, F.-J., Analysis of HDG methods for Stokes flow, Math. Comput., 80, 723-760 (2011) · Zbl 1410.76164 |
| [16] | Cockburn, B.; Gopalakrishnan, J.; Sayas, F.-J., A projection-based error analysis of HDG methods, Math. Comput., 79, 1351-1367 (2010) · Zbl 1197.65173 |
| [17] | Cockburn, B.; Guzmán, J.; Soon, S.-C.; Stolarski, H. K., An analysis of the embedded discontinuous Galerkin method for second-order elliptic problems, SIAM J. Numer. Anal., 47, 2686-2707 (2009) · Zbl 1211.65153 |
| [18] | Cockburn, B.; Guzmán, J.; Wang, H., Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. Comput., 78, 1-24 (2009) · Zbl 1198.65194 |
| [19] | Cockburn, B.; Kanschat, G.; Schötzau, D.; Schwab, C., Local discontinuous Galerkin methods for the Stokes system, SIAM J. Numer. Anal., 40, 319-343 (2002) · Zbl 1032.65127 |
| [20] | Cockburn, B.; Kanschat, G.; Schötzau, D., Local discontinuous Galerkin methods for the Oseen equations, Math. Comput., 73, 569-593 (2004) · Zbl 1066.76036 |
| [21] | Cockburn, B.; Kanschat, G.; Schötzau, D., A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comput., 74, 1067-1095 (2005) · Zbl 1069.76029 |
| [22] | Cockburn, B.; Kanschat, G.; Schötzau, D., A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations, J. Sci. Comput., 31, 61-73 (2007) · Zbl 1151.76527 |
| [23] | Cockburn, B.; Kanschat, G.; Schötzau, D., An equal-order DG method for the incompressible Navier-Stokes equations, J. Sci. Comput., 40, 141-187 (2009) |
| [24] | Cockburn, B.; Qiu, W.; Shi, K., Conditions for superconvergence of HDG methods for second-order elliptic problems, Math. Comput., 81, 1327-1353 (2012) · Zbl 1251.65158 |
| [25] | Cockburn, B.; Qui, W.; Shi, K., Conditions for superconvergence of HDG methods on curvilinear elements for second-order elliptic problems, SIAM J. Numer. Anal., 50, 1417-1432 (2012) · Zbl 1256.65093 |
| [26] | Cockburn, B.; Shi, K., Conditions for superconvergence of HDG methods for Stokes flow, Math. Comput., 82, 651-671 (2013) · Zbl 1322.65104 |
| [27] | Cockburn, B.; Shi, K., Superconvergent HDG methods for linear elasticity with weakly symmetric stresses, IMA J. Numer. Anal., 33, 747-770 (2013) · Zbl 1441.74235 |
| [28] | Cockburn, B.; Shu, C. W., The local discontinuous Galerkin method for convection-diffusion systems, SIAM J. Numer. Anal., 35, 2440-2463 (1998) · Zbl 0927.65118 |
| [29] | Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 3, 173-261 (2001) · Zbl 1065.76135 |
| [30] | Demkowicz, L.; Gopalakrishnan, J., A class of discontinuous Petrov-Galerkin methods. Part I: The transport equation, Comput. Methods Appl. Mech. Eng., 199, 1558-1572 (2010) · Zbl 1231.76142 |
| [31] | Fidkowski, K. J.; Oliver, T. A.; Lu, J.; Darmofal, D. L., \(p\)-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations, J. Comput. Phys., 207, 1, 92-113 (2005) · Zbl 1177.76194 |
| [32] | Güzey, S.; Cockburn, B.; Stolarski, H., The embedded discontinuous Galerkin methods: application to linear shell problems, Int. J. Numer. Methods Eng., 70, 757-790 (2007) · Zbl 1194.74403 |
| [33] | Kirby, R. M.; Sherwin, S. J.; Cockburn, B., To HDG or to CG: a comparative study, J. Sci. Comput., 51, 183-212 (2012) · Zbl 1244.65174 |
| [34] | Hartmann, R.; Houston, P., Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations, J. Comput. Phys., 183, 508-532 (2002) · Zbl 1057.76033 |
| [35] | Hesthaven, J. S.; Warburton, T., Nodal high-order methods on unstructured grids, I. Time-domain solution of Maxwell’s equations, J. Comput. Phys., 181, 1, 186-221 (2002) · Zbl 1014.78016 |
| [36] | Huerta, A.; Angeloski, A.; Roca, X.; Peraire, J., Efficiency of high-order elements for continuous and discontinuous Galerkin methods, Int. J. Numer. Methods Eng., 96, 529-560 (2013) · Zbl 1352.65512 |
| [37] | Hughes, T. J.R.; Scovazzi, G.; Bochev, P. B.; Buffa, A., A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method, Comput. Methods Appl. Mech. Eng., 195, 2761-2787 (2006) · Zbl 1124.76027 |
| [38] | Klaij, C. M.; van der Vegt, J. J.W.; van der Ven, H., Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations, J. Comput. Phys., 217, 2, 589-611 (2006) · Zbl 1099.76035 |
| [39] | Lomtev, I.; Karniadakis, G. E., A discontinuous Galerkin method for the Navier-Stokes equations, Int. J. Numer. Methods Fluids, 29, 587-603 (1999) · Zbl 0951.76041 |
| [40] | Lomtev, I.; Kirby, R. M.; Karniadakis, G. E., A discontinuous Galerkin ALE method for compressible viscous flows in moving domains, J. Comput. Phys., 155, 1, 128-159 (1999) · Zbl 0956.76046 |
| [41] | Montlaur, A.; Fernández-Méndez, S.; Peraire, J.; Huerta, A., Discontinuous Galerkin methods for the Navier-Stokes equations using solenoidal approximations, Int. J. Numer. Methods Fluids, 64, 549-564 (2010) · Zbl 1377.76008 |
| [42] | Moro, D.; Nguyen, N. C.; Peraire, J.; Gopalakrishnan, J., A hybridized discontinuous Petrov-Galerkin method for compressible flows, (Proceedings of the 49th AIAA Aerospace Sciences Meeting and Exhibit. Proceedings of the 49th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, FL (January 2011)), (AIAA Paper 2011-197) |
| [43] | Moro, D.; Nguyen, N. C.; Peraire, J., A hybridized discontinuous Petrov-Galerkin scheme for scalar conservation laws, Int. J. Numer. Methods Eng., 91, 950-970 (2012) |
| [44] | Moro, D.; Nguyen, N. C.; Peraire, J., Navier-Stokes solution using hybridizable discontinuous Galerkin methods, (Proceedings of the 20th AIAA Computational Fluid Dynamics Conference. Proceedings of the 20th AIAA Computational Fluid Dynamics Conference, Honolulu, HI (June 2011)), (AIAA Paper 2011-3060) |
| [45] | Nguyen, N. C.; Peraire, J., Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics, J. Comput. Phys., 231, 5955-5988 (2012) · Zbl 1277.65082 |
| [46] | Nguyen, N. C.; Peraire, J., An adaptive shock-capturing HDG method for compressible flows, (Proceedings of the 20th AIAA Computational Fluid Dynamics Conference. Proceedings of the 20th AIAA Computational Fluid Dynamics Conference, Honolulu, HI (June 2011)), (AIAA Paper 2011-3407) |
| [47] | Nguyen, N. C.; Peraire, J.; Cockburn, B., A comparison of HDG methods for Stokes flow, J. Sci. Comput., 45, 1-3, 215-237 (2010) · Zbl 1203.76079 |
| [48] | Nguyen, N. C.; Peraire, J.; Cockburn, B., Implicit high-order hybridizable discontinuous Galerkin methods for acoustics and elastodynamics, J. Comput. Phys., 230, 3695-3718 (2011) · Zbl 1364.76093 |
| [49] | Nguyen, N. C.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations, J. Comput. Phys., 230, 1147-1170 (2011) · Zbl 1391.76353 |
| [50] | Nguyen, N. C.; Peraire, J.; Cockburn, B., Hybridizable discontinuous Galerkin methods, (Hesthaven, J. S.; Ronquist, E. M., Spectral and High Order Methods for Partial Differential Equations. Spectral and High Order Methods for Partial Differential Equations, Lect. Notes Comput. Sci. Eng., vol. 76 (2011)), 63-84 · Zbl 1216.65160 |
| [51] | Nguyen, N. C.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, J. Comput. Phys., 228, 3232-3254 (2009) · Zbl 1187.65110 |
| [52] | Nguyen, N. C.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations, J. Comput. Phys., 228, 8841-8855 (2009) · Zbl 1177.65150 |
| [53] | Nguyen, N. C.; Peraire, J.; Cockburn, B., A hybridizable discontinuous Galerkin method for Stokes flow, Comput. Methods Appl. Mech. Eng., 199, 582-597 (2010) · Zbl 1227.76036 |
| [54] | Nguyen, N. C.; Peraire, J.; Cockburn, B., A hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations, (Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit. Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, FL (January 2010)), (AIAA Paper 2010-362) · Zbl 1227.76036 |
| [55] | Peraire, J.; Nguyen, N. C.; Cockburn, B., An embedded discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations, (Proceedings of the 20th AIAA Computational Fluid Dynamics Conference. Proceedings of the 20th AIAA Computational Fluid Dynamics Conference, Honolulu, HI (2011)), (AIAA Paper 2011-3228) · Zbl 1391.76353 |
| [56] | Peraire, J.; Nguyen, N. C.; Cockburn, B., A hybridizable discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations, (Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit. Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, FL (January 2010)), (AIAA Paper 2010-363) · Zbl 1227.76036 |
| [57] | Peraire, J.; Persson, P. O., The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J. Sci. Comput., 30, 4, 1806-1824 (2008) · Zbl 1167.65436 |
| [58] | Persson, P. O.; Peraire, J., Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the Navier-Stokes equations, SIAM J. Sci. Comput., 30, 6, 2709-2733 (2008) · Zbl 1362.76052 |
| [59] | Persson, P.-O.; Bonet, J.; Peraire, J., Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains, Comput. Methods Appl. Mech. Eng., 198, 1585-1595 (2009) · Zbl 1227.76038 |
| [60] | Raviart, P. A.; Thomas, J. M., A mixed finite element method for second order elliptic problems, (Galligani, I.; Magenes, E., Mathematical Aspects of Finite Element Method. Mathematical Aspects of Finite Element Method, Lect. Notes Math., vol. 606 (1977), Springer-Verlag: Springer-Verlag New York), 292-315 · Zbl 0362.65089 |
| [61] | Reed, W. H.; Hill, T. R., Triangular mesh methods for the neutron transport equation (1973), Los Alamos Scientific Laboratory, Technical Report LA-UR-73-479 |
| [62] | Roca, X.; Peraire, J.; Nguyen, N. C., Scalable parallelization of the hybridized discontinuous Galerkin method for compressible flow, (Proceedings of the 21th AIAA Computational Fluid Dynamics Conference. Proceedings of the 21th AIAA Computational Fluid Dynamics Conference, San Diego, CA (June 2013)), (AIAA Paper 2013-2939) |
| [63] | Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372 (1981) · Zbl 0474.65066 |
| [64] | Shahbazi, K.; Fischer, P. F.; Ethier, C. R., A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations, J. Comput. Phys., 222, 391-407 (2007) · Zbl 1216.76034 |
| [65] | Ueckermann, M. P.; Lermusiaux, P. F.J., High order schemes for 2D unsteady biogeochemical ocean models, Ocean Dyn., 60, 1415-1445 (2010) |
| [66] | Wang, Z. J.; Fidkowski, K.; Abgrall, R.; Bassi, F.; Caraeni, D.; Cary, A.; Deconinck, H.; Hartmann, R.; Hillewaert, K.; Huynh, H. T.; Kroll, N.; May, G.; Persson, P.-O.; van Leer, B.; Visbal, M., High-order CFD methods: current status and perspective, Int. J. Numer. Methods Fluids, 72, 811-845 (2013) · Zbl 1455.76007 |
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