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Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. (English) Zbl 1380.65253
Summary: It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws [G. Jiang and the second author, Math. Comput. 62, No. 206, 531–538 (1994; Zbl 0801.65098)] and symmetric hyperbolic systems [S. Hou and X.-D. Liu, J. Sci. Comput. 31, No. 1–2, 127–151 (2007; Zbl 1152.76433)], in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and/or when the integration is approximated by a numerical quadrature. In this paper, we develop a unified framework for designing high order DG methods which will satisfy entropy inequalities for any given single convex entropy, through suitable numerical quadrature which is specific to this given entropy. Our framework applies from one-dimensional scalar cases all the way to multi-dimensional systems of conservation laws. For the one-dimensional case, our numerical quadrature is based on the methodology established in [M. H. Carpenter et al., SIAM J. Sci. Comput. 36, No. 5, B835–B867 (2014; Zbl 1457.65140); G. J. Gassner, SIAM J. Sci. Comput. 35, No. 3, A1233–A1253 (2013; Zbl 1275.65065)]. The main ingredients are summation-by-parts (SBP) operators derived from Legendre Gauss-Lobatto quadrature, the entropy conservative flux within elements, and the entropy stable flux at element interfaces. We then generalize the scheme to two-dimensional triangular meshes by constructing SBP operators on triangles based on a special quadrature rule. A local discontinuous Galerkin (LDG) type treatment is also incorporated to achieve the generalization to convection-diffusion equations. Extensive numerical experiments are performed to validate the accuracy and shock capturing efficacy of these entropy stable DG methods.

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35D30 Weak solutions to PDEs
35R45 Partial differential inequalities and systems of partial differential inequalities
Full Text: DOI
[1] Barth, T. J., Numerical methods for gasdynamic systems on unstructured meshes, (An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, Lecture Notes in Computational Science and Engineering, vol. 5, (1999), Springer), 195-285 · Zbl 0969.76040
[2] Beirão da Veiga, L.; Brezzi, F.; Cangiani, A.; Manzini, G.; Marini, L.; Russo, A., Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23, 199-214, (2013) · Zbl 1416.65433
[3] Beirão da Veiga, L.; Brezzi, F.; Marini, L.; Russo, A., The Hitchhiker’s guide to the virtual element method, Math. Models Methods Appl. Sci., 24, 1541-1573, (2014) · Zbl 1291.65336
[4] Bouchut, F.; Bourdarias, C.; Perthame, B., A MUSCL method satisfying all the numerical entropy inequalities, Math. Comput., 65, 1439-1461, (1996) · Zbl 0853.65091
[5] Carpenter, M. H.; Fisher, T. C.; Nielsen, E. J.; Frankel, S. H., Entropy stable spectral collocation schemes for the Navier-Stokes equations: discontinuous interfaces, SIAM J. Sci. Comput., 36, B835-B867, (2014) · Zbl 1457.65140
[6] Castillo, P.; Cockburn, B.; Perugia, I.; Schötzau, D., An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal., 38, 1676-1706, (2000) · Zbl 0987.65111
[7] Chandrashekar, P., Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations, Commun. Comput. Phys., 14, 1252-1286, (2013) · Zbl 1373.76121
[8] Cockburn, B.; Hou, S.; Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV: the multidimensional case, Math. Comput., 54, 545-581, (1990) · Zbl 0695.65066
[9] 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, J. Comput. Phys., 84, 90-113, (1989) · Zbl 0677.65093
[10] Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II: general framework, Math. Comput., 52, 411-435, (1989) · Zbl 0662.65083
[11] Cockburn, B.; Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35, 2440-2463, (1998) · Zbl 0927.65118
[12] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws. V: multidimensional systems, J. Comput. Phys., 141, 199-224, (1998) · Zbl 0920.65059
[13] Crandall, M. G.; Majda, A., Monotone difference approximations for scalar conservation laws, Math. Comput., 34, 1-21, (1980) · Zbl 0423.65052
[14] Dafermos, C. M., Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften, vol. 325, (2010), Springer · Zbl 1196.35001
[15] Fisher, T. C.; Carpenter, M. H., High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains, J. Comput. Phys., 252, 518-557, (2013) · Zbl 1349.65293
[16] Fisher, T. C.; Carpenter, M. H.; Nordström, J.; Yamaleev, N. K.; Swanson, C., Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: theory and boundary conditions, J. Comput. Phys., 234, 353-375, (2013) · Zbl 1284.65102
[17] Fjordholm, U. S.; Mishra, S.; Tadmor, E., Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws, SIAM J. Numer. Anal., 50, 544-573, (2012) · Zbl 1252.65150
[18] Fjordholm, U. S.; Mishra, S.; Tadmor, E., ENO reconstruction and ENO interpolation are stable, Found. Comput. Math., 13, 139-159, (2013) · Zbl 1273.65120
[19] Gassner, G. J., A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods, SIAM J. Sci. Comput., 35, A1233-A1253, (2013) · Zbl 1275.65065
[20] Gassner, G. J.; Winters, A. R.; Kopriva, D. A., A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations, Appl. Math. Comput., 272, 291-308, (2016)
[21] Geuzaine, C.; Remacle, J.-F., Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Methods Eng., 79, 1309-1331, (2009) · Zbl 1176.74181
[22] Godlewski, E.; Raviart, P.-A., Hyperbolic systems of conservation laws, Mathématiques & Applications, (1991), Ellipsis · Zbl 0768.35059
[23] Godlewski, E.; Raviart, P.-A., Numerical approximation of hyperbolic systems of conservation laws, Appl. Math. Sci., vol. 118, (2013), Springer · Zbl 1063.65080
[24] Godunov, S. K., An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR, 139, 521-523, (1961) · Zbl 0125.06002
[25] Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev., 43, 89-112, (2001) · Zbl 0967.65098
[26] Guermond, J.-L.; Pasquetti, R.; Popov, B., Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230, 4248-4267, (2011) · Zbl 1220.65134
[27] Guermond, J.-L.; Popov, B., Fast estimation from above of the maximum wave speed in the Riemann problem for the Euler equations, J. Comput. Phys., 321, 908-926, (2016) · Zbl 1349.76769
[28] Gustafsson, B.; Kreiss, H.-O.; Oliger, J., Time dependent problems and difference methods, (1995), John Wiley & Sons
[29] Harten, A., On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys., 49, 151-164, (1983) · Zbl 0503.76088
[30] Harten, A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49, 357-393, (1983) · Zbl 0565.65050
[31] Harten, A.; Hyman, J. M.; Lax, P. D.; Keyfitz, B., On finite-difference approximations and entropy conditions for shocks, Commun. Pure Appl. Math., 29, 297-322, (1976) · Zbl 0351.76070
[32] Harten, A.; Lax, P. D.; Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 35-61, (1983) · Zbl 0565.65051
[33] Hesthaven, J. S.; Warburton, T., Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, (2007), Springer
[34] Hicken, J. E.; Del Rey Fernández, D. C.; Zingg, D. W., Multidimensional summation-by-parts operators: general theory and application to simplex elements, SIAM J. Sci. Comput., 38, A1935-A1958, (2016) · Zbl 1382.65355
[35] Hiltebrand, A.; Mishra, S., Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws, Numer. Math., 126, 103-151, (2014) · Zbl 1303.65083
[36] Hou, S.; Liu, X.-D., Solutions of multi-dimensional hyperbolic systems of conservation laws by square entropy condition satisfying discontinuous Galerkin method, J. Sci. Comput., 31, 127-151, (2007) · Zbl 1152.76433
[37] Hughes, T. J.; Franca, L.; Mallet, M., 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. Eng., 54, 223-234, (1986) · Zbl 0572.76068
[38] Ismail, F.; Roe, P. L., Affordable, entropy-consistent Euler flux functions II: entropy production at shocks, J. Comput. Phys., 228, 5410-5436, (2009) · Zbl 1280.76015
[39] Jiang, G. S.; Shu, C.-W., On a cell entropy inequality for discontinuous Galerkin methods, Math. Comput., 62, 531-538, (1994) · Zbl 0801.65098
[40] Karniadakis, G.; Sherwin, S., Spectral/hp element methods for computational fluid dynamics, (2013), Oxford University Press · Zbl 1256.76003
[41] Kopriva, D. A.; Gassner, G., On the quadrature and weak form choices in collocation type discontinuous Galerkin spectral element methods, J. Sci. Comput., 44, 136-155, (2010) · Zbl 1203.65199
[42] Kruzhkov, S. N., First order quasilinear equations in several independent variables, Mat. Sb., 123, 228-255, (1970) · Zbl 0202.11203
[43] Lax, P.; Wendroff, B., Systems of conservation laws, Commun. Pure Appl. Math., 13, 217-237, (1960) · Zbl 0152.44802
[44] Lefloch, P. G.; Mercier, J.-M.; Rohde, C., Fully discrete, entropy conservative schemes of arbitrary order, SIAM J. Numer. Anal., 40, 1968-1992, (2002) · Zbl 1033.65073
[45] Mock, M. S., Systems of conservation laws of mixed type, J. Differ. Equ., 37, 70-88, (1980) · Zbl 0413.34017
[46] Osher, S., Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal., 21, 217-235, (1984) · Zbl 0592.65069
[47] Osher, S.; Tadmor, E., On the convergence of difference approximations to scalar conservation laws, Math. Comput., 50, 19-51, (1988) · Zbl 0637.65091
[48] Panov, E. Y., Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Math. Notes, 55, 517-525, (1994)
[49] Ray, D.; Chandrashekar, P.; Fjordholm, U. S.; Mishra, S., Entropy stable scheme on two-dimensional unstructured grids for Euler equations, Commun. Comput. Phys., 19, 1111-1140, (2016) · Zbl 1373.76143
[50] Shu, C.-W., TVB uniformly high-order schemes for conservation laws, Math. Comput., 49, 105-121, (1987) · Zbl 0628.65075
[51] Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, (Cockburn, B.; Johnson, C.; Shu, C.-W.; Tadmor, E.; Quarteroni, A., Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, (1998), Springer Berlin), 325-432 · Zbl 0927.65111
[52] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439-471, (1988) · Zbl 0653.65072
[53] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. Comput. Phys., 83, 32-78, (1989) · Zbl 0674.65061
[54] Svärd, M.; Özcan, H., Entropy-stable schemes for the Euler equations with far-field and wall boundary conditions, J. Sci. Comput., 58, 61-89, (2014) · Zbl 1290.65084
[55] Tadmor, E., Skew-selfadjoint form for systems of conservation laws, J. Math. Anal. Appl., 103, 428-442, (1984) · Zbl 0599.35102
[56] Tadmor, E., The numerical viscosity of entropy stable schemes for systems of conservation laws. I, Math. Comput., 49, 91-103, (1987) · Zbl 0641.65068
[57] Tadmor, E., Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta Numer., 12, 451-512, (2003) · Zbl 1046.65078
[58] Tan, S.; Shu, C.-W., Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws, J. Comput. Phys., 229, 8144-8166, (2010) · Zbl 1198.65174
[59] Toro, E. F., Shock-capturing methods for free-surface shallow flows, (2001), John Wiley · Zbl 0996.76003
[60] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics: A practical introduction, (2013), Springer
[61] Wagner, D., The Riemann problem in two space dimensions for a single conservation law, SIAM J. Math. Anal., 14, 534-559, (1983) · Zbl 0526.35052
[62] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 115-173, (1984) · Zbl 0573.76057
[63] Zhang, L.; Cui, T.; Liu, H., A set of symmetric quadrature rules on triangles and tetrahedra, J. Comput. Math., 27, 89-96, (2009) · Zbl 1199.65081
[64] Zhang, Q.; Shu, C.-W., Stability analysis and a priori error estimates of the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws, SIAM J. Numer. Anal., 48, 1038-1063, (2010) · Zbl 1217.65178
[65] Zhang, X., On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier-Stokes equations, J. Comput. Phys., 328, 301-343, (2017)
[66] Zhang, X.; Shu, C.-W., On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys., 229, 3091-3120, (2010) · Zbl 1187.65096
[67] Zhang, X.; Shu, C.-W., On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys., 229, 8918-8934, (2010) · Zbl 1282.76128
[68] Zhang, X.; Xia, Y.; Shu, C.-W., Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes, J. Sci. Comput., 50, 29-62, (2012) · Zbl 1247.65131
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