zbMATH — the first resource for mathematics

On discretely entropy conservative and entropy stable discontinuous Galerkin methods. (English) Zbl 1391.76310
Summary: High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matrices. In this work, we describe how use flux differencing, quadrature-based projections, and SBP-like operators to construct discretely entropy conservative schemes for DG methods under more arbitrary choices of volume and surface quadrature rules. The resulting methods are semi-discretely entropy conservative or entropy stable with respect to the volume quadrature rule used. Numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible Euler equations in one and two dimensions.

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
35L65 Hyperbolic conservation laws
Full Text: DOI arXiv
[1] Fisher, Travis C.; Carpenter, Mark H., High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains, J. Comput. Phys., 252, 518-557, (2013) · Zbl 1349.65293
[2] Carpenter, Mark H.; Fisher, Travis C.; Nielsen, Eric J.; Frankel, Steven H., Entropy stable spectral collocation schemes for the Navier-Stokes equations: discontinuous interfaces, SIAM J. Sci. Comput., 36, 5, B835-B867, (2014) · Zbl 1457.65140
[3] Gassner, Gregor J.; Winters, Andrew R.; Kopriva, David A., Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations, J. Comput. Phys., 327, 39-66, (2016) · Zbl 1422.65280
[4] Gassner, Gregor J.; Winters, Andrew R.; Kopriva, David A., A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations, Appl. Math. Comput., 272, 291-308, (2016) · Zbl 1410.65393
[5] Wintermeyer, Niklas; Winters, Andrew R.; Gassner, Gregor J.; Kopriva, David A., An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry, J. Comput. Phys., 340, 200-242, (2017) · Zbl 1380.65291
[6] Chen, Tianheng; Shu, Chi-Wang, Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws, J. Comput. Phys., 345, 427-461, (2017) · Zbl 1380.65253
[7] Hesthaven, Jan S.; Warburton, Tim, Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, vol. 54, (2007), Springer · Zbl 1134.65068
[8] Klöckner, Andreas; Warburton, Tim; Bridge, Jeff; Hesthaven, Jan S., Nodal discontinuous Galerkin methods on graphics processors, J. Comput. Phys., 228, 21, 7863-7882, (2009) · Zbl 1175.65111
[9] Ainsworth, Mark, Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods, J. Comput. Phys., 198, 1, 106-130, (2004) · Zbl 1058.65103
[10] Wilcox, Lucas C.; Stadler, Georg; Burstedde, Carsten; Ghattas, Omar, A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media, J. Comput. Phys., 229, 24, 9373-9396, (2010) · Zbl 1427.74071
[11] Wang, Zhijian J.; Fidkowski, Krzysztof; Abgrall, Rémi; Bassi, Francesco; Caraeni, Doru; Cary, Andrew; Deconinck, Herman; Hartmann, Ralf; Hillewaert, Koen; Huynh, Hung T., High-order CFD methods: current status and perspective, Int. J. Numer. Methods Fluids, 72, 8, 811-845, (2013)
[12] Krivodonova, Lilia, Limiters for high-order discontinuous Galerkin methods, J. Comput. Phys., 226, 1, 879-896, (2007) · Zbl 1125.65091
[13] Persson, Per-Olof; Peraire, Jaime, Sub-cell shock capturing for discontinuous Galerkin methods, (AIAA, vol. 112, (2006)) · Zbl 1358.76043
[14] Kirby, Robert M.; Em Karniadakis, George, De-aliasing on non-uniform grids: algorithms and applications, J. Comput. Phys., 191, 1, 249-264, (2003) · Zbl 1161.76534
[15] Warburton, T., A low-storage curvilinear discontinuous Galerkin method for wave problems, SIAM J. Sci. Comput., 35, 4, A1987-A2012, (2013) · Zbl 1362.65109
[16] Chan, Jesse; Hewett, Russell J.; Warburton, T., Weight-adjusted discontinuous Galerkin methods: wave propagation in heterogeneous media, (2016), arXiv preprint · Zbl 1379.65075
[17] Chan, Jesse; Hewett, Russell J.; Warburton, T., Weight-adjusted discontinuous Galerkin methods: curvilinear meshes, (2016), arXiv preprint · Zbl 1377.65127
[18] Chan, Jesse, Weight-adjusted discontinuous Galerkin methods: matrix-valued weights and elastic wave propagation in heterogeneous media, (2017), arXiv preprint · Zbl 1379.65075
[19] Tadmor, Eitan, The numerical viscosity of entropy stable schemes for systems of conservation laws. I, Math. Comput., 49, 179, 91-103, (1987) · Zbl 0641.65068
[20] Ray, Deep; Chandrashekar, Praveen; Fjordholm, Ulrik S.; Mishra, Siddhartha, Entropy stable scheme on two-dimensional unstructured grids for Euler equations, Commun. Comput. Phys., 19, 5, 1111-1140, (2016) · Zbl 1373.76143
[21] Fjordholm, Ulrik S.; Mishra, Siddhartha; Tadmor, Eitan, Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws, SIAM J. Numer. Anal., 50, 2, 544-573, (2012) · Zbl 1252.65150
[22] Gassner, Gregor J., A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods, SIAM J. Sci. Comput., 35, 3, A1233-A1253, (2013) · Zbl 1275.65065
[23] Winters, Andrew R.; Derigs, Dominik; Gassner, Gregor J.; Walch, Stefanie, A uniquely defined entropy stable matrix dissipation operator for high Mach number ideal MHD and compressible Euler simulations, J. Comput. Phys., 332, 274-289, (2017) · Zbl 1378.76144
[24] Crean, Jared; Hicken, Jason E.; Del Rey Fernández, David C.; Zingg, David W.; Carpenter, Mark H., High-order entropy-stable discretizations of the Euler equations for complex geometries, (23rd AIAA Computational Fluid Dynamics Conference, (2017), American Institute of Aeronautics and Astronautics) · Zbl 1380.76080
[25] Gassner, Gregor J., A kinetic energy preserving nodal discontinuous Galerkin spectral element method, Int. J. Numer. Methods Fluids, 76, 1, 28-50, (2014)
[26] Ortleb, Sigrun, Kinetic energy preserving DG schemes based on summation-by-parts operators on interior node distributions, PAMM, 16, 1, 857-858, (2016)
[27] Ortleb, Sigrun, A kinetic energy preserving DG scheme based on Gauss-Legendre points, J. Sci. Comput., 71, 3, 1135-1168, (2017) · Zbl 06759489
[28] Ranocha, Hendrik; Öffner, Philipp; Sonar, Thomas, Extended skew-symmetric form for summation-by-parts operators and varying Jacobians, J. Comput. Phys., 342, 13-28, (2017) · Zbl 1380.65318
[29] Ranocha, Hendrik, Generalised summation-by-parts operators and variable coefficients, (2017), arXiv preprint · Zbl 1395.65054
[30] Hughes, Thomas J. R.; Franca, L. P.; 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, 2, 223-234, (1986) · Zbl 0572.76068
[31] Chin-Joe-Kong, M. J.S.; Mulder, W. A.; Van Veldhuizen, M., Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation, J. Eng. Math., 35, 4, 405-426, (1999) · Zbl 0948.74057
[32] Hicken, Jason E.; Del Rey Fernández, David C.; Zingg, David W., Multidimensional summation-by-parts operators: general theory and application to simplex elements, SIAM J. Sci. Comput., 38, 4, A1935-A1958, (2016) · Zbl 1382.65355
[33] Zhebel, Elena; Minisini, Sara; Kononov, Alexey; Mulder, Wim A., A comparison of continuous mass-lumped finite elements with finite differences for 3-D wave propagation, Geophys. Prospect., 62, 5, 1111-1125, (2014)
[34] Chan, Jesse; Warburton, T., Orthogonal bases for vertex-mapped pyramids, SIAM J. Sci. Comput., 38, 2, A1146-A1170, (2016) · Zbl 1416.65438
[35] Xiao, H.; Gimbutas, Zydrunas, A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions, Comput. Math. Appl., 59, 663-676, (2010) · Zbl 1189.65047
[36] Del Rey Fernández, David C.; Boom, Pieter D.; Zingg, David W., A generalized framework for nodal first derivative summation-by-parts operators, J. Comput. Phys., 266, 214-239, (2014) · Zbl 1311.65002
[37] Ranocha, Hendrik, Comparison of some entropy conservative numerical fluxes for the Euler equations, (2017), arXiv preprint · Zbl 1432.65156
[38] Mock, Michael S., Systems of conservation laws of mixed type, J. Differ. Equ., 37, 1, 70-88, (1980) · Zbl 0413.34017
[39] Chan, Jesse; Wang, Zheng; Modave, Axel; Remacle, Jean-Francois; Warburton, T., GPU-accelerated discontinuous Galerkin methods on hybrid meshes, J. Comput. Phys., 318, 142-168, (2016) · Zbl 1349.65443
[40] Di Pietro, Daniele Antonio; Ern, Alexandre, Mathematical aspects of discontinuous Galerkin methods, vol. 69, (2011), Springer Science & Business Media · Zbl 1231.65209
[41] Gassner, Gregor J.; Winters, Andrew R.; Hindenlang, Florian J.; Kopriva, David A., The BR1 scheme is stable for the compressible Navier-Stokes equations, (2017), arXiv preprint · Zbl 06993418
[42] Crean, Jared; Hicken, Jason E.; Del Rey Fernández, David C.; Zingg, David W.; Carpenter, Mark H., Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements, J. Comput. Phys., 356, 410-438, (2018) · Zbl 1380.76080
[43] Horn, Roger A.; Johnson, Charles R., Matrix analysis, (2012), Cambridge University Press · Zbl 0704.15002
[44] Parsani, Matteo; Carpenter, Mark H.; Fisher, Travis C.; Nielsen, Eric J., Entropy stable staggered grid discontinuous spectral collocation methods of any order for the compressible Navier-Stokes equations, SIAM J. Sci. Comput., 38, 5, A3129-A3162, (2016) · Zbl 1457.65149
[45] Zhang, Xiangxiong; Shu, Chi-Wang, On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys., 229, 23, 8918-8934, (2010) · Zbl 1282.76128
[46] Zhang, Xiangxiong; Xia, Yinhua; Shu, Chi-Wang, Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes, J. Sci. Comput., 50, 1, 29-62, (2012) · Zbl 1247.65131
[47] Shi, Cengke; Shu, Chi-Wang, On local conservation of numerical methods for conservation laws, Comput. Fluids, (2017), in press · Zbl 1410.65327
[48] Chandrashekar, Praveen, Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations, Commun. Comput. Phys., 14, 5, 1252-1286, (2013) · Zbl 1373.76121
[49] Magnus, Svärd; Özcan, Hatice, Entropy-stable schemes for the Euler equations with far-field and wall boundary conditions, J. Sci. Comput., 58, 1, 61-89, (2014) · Zbl 1290.65084
[50] Parsani, Matteo; Carpenter, Mark H.; Nielsen, Eric J., Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations, J. Comput. Phys., 292, 88-113, (2015) · Zbl 1349.76639
[51] Carpenter, Mark H.; Kennedy, Christopher A., Fourth-order 2n-storage Runge-Kutta schemes, (1994), NASA Langley Research Center, Technical Report NASA-TM-109112, NAS 1.15:109112
[52] Warburton, T.; Hesthaven, Jan S., On the constants in hp-finite element trace inverse inequalities, Comput. Methods Appl. Mech. Eng., 192, 25, 2765-2773, (2003) · Zbl 1038.65116
[53] Harten, Amiram, On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys., 49, 1, 151-164, (1983) · Zbl 0503.76088
[54] Ismail, Farzad; Roe, Philip L., Affordable, entropy-consistent Euler flux functions II: entropy production at shocks, J. Comput. Phys., 228, 15, 5410-5436, (2009) · Zbl 1280.76015
[55] Shu, Chi-Wang, High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev., 51, 1, 82-126, (2009) · Zbl 1160.65330
[56] Shu, Chi-Wang, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, (Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, (1998), Springer), 325-432 · Zbl 0927.65111
[57] Johnson, Claes; Pitkäranta, Juhani, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comput., 46, 173, 1-26, (1986) · Zbl 0618.65105
[58] Cockburn, Bernardo; Dong, Bo; Guzmán, Johnny, Optimal convergence of the original DG method for the transport-reaction equation on special meshes, SIAM J. Numer. Anal., 46, 3, 1250-1265, (2008) · Zbl 1168.65058
[59] Lax, Peter D.; Liu, Xu-Dong, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19, 2, 319-340, (1998) · Zbl 0952.76060
[60] Kurganov, Alexander; Tadmor, Eitan, Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numer. Methods Partial Differ. Equ., 18, 5, 584-608, (2002) · Zbl 1058.76046
[61] Liska, Richard; Wendroff, Burton, Comparison of several difference schemes on 1D and 2D test problems for the Euler equations, SIAM J. Sci. Comput., 25, 3, 995-1017, (2003) · Zbl 1096.65089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.