Review of Entropy Stable Discontinuous Galerkin Methods for Systems of Conservation Laws on Unstructured Simplex Meshes

Abstract

In this paper, we will build a roadmap for the growing literature of high order quadrature-based entropy stable discontinuous Galerkin (DG) methods, trying to elucidate the motivations and emphasize the contributions. Compared to the classic DG method which is only provably stable for the square entropy, these DG methods can be tailored to satisfy an arbitrary given entropy inequality, and do not require exact integration. The methodology is within the summation-by-parts (SBP) paradigm, such that the discrete operators collocated at the quadrature points should satisfy the SBP property. The construction is relatively easy for quadrature rules with collocated surface nodes. We use the flux differencing technique to ensure entropy balance within elements, and the simultaneous approximation terms (SATs) to produce entropy dis-sipation on element interfaces. The further extension to general quadrature rules is achieved through careful modifications of SATs.

Keywords

System of conservation laws / entropy stability / discontinuous Galerkin method / summation-by-parts

References

[1] R. Abgrall. A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes. Journal of Computational Physics, 372:640-666, 2018.

[2] T. J. Barth. Numerical methods for gasdynamic systems on unstructured meshes. In An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, Springer, 1999, pp. 195-285.

[3] M. W. Bohm, A. R. Winters, G. J. Gassner, D. Derigs, F. J. Hindenlang and J. Saur. An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: The-ory and numerical verification. Journal of Computational Physics, Article 108076, 2018. DOI: 10.1016/j.jcp.2018.06.027

[4] M. H. Carpenter, T. C. Fisher, E. J. Nielsen, M. Parsani, M. Svärd and N. Yamaleev. Entropy stable summation-by-parts formulations for compressible computational fluid dynamics. In Handbook of Numerical Analysis, volume 17, Elsevier, 2016, pp. 495-524.

[5] M. H. Carpenter, T. C. Fisher, E. J. Nielsen and S. H. Frankel. Entropy stable spectral collo-cation schemes for the Navier-Stokes equations: discontinuous interfaces. SIAM Journal on Scientific Computing, 36:B835-B867, 2014.

[6] P. Castillo, B. Cockburn, I. Perugia, and D. Schötzau. An a priori error analysis of the local discontinuous Galerkin method for elliptic problems SIAM Journal on Numerical Analysis, 38:1676-1706, 2000.

[7] J. Chan. On discretely entropy conservative and entropy stable discontinuous Galerkin methods. Journal of Computational Physics, 362:346-374, 2018.

[8] J. Chan, D. C. Del Rey Fernández and M. H. Carpenter. Efficient entropy stable Gauss collo-cation methods. SIAM Journal on Scientific Computing, 41:A2938-A2966, 2019.

[9] J. Chan and L. C. Wilcox. On discretely entropy stable weight-adjusted discontinuous Galerkin methods: curvilinear meshes. Journal of Computational Physics, 378:366-393, 2019.

[10] P. Chandrashekar. Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations. Communications in Computational Physics, 14:1252-1286, 2013.

[11] P. Chandrashekar and C. Klingenberg. Entropy stable finite volume scheme for ideal com-pressible MHD on 2-D Cartesian meshes. SIAM Journal on Numerical Analysis, 54:1313-1340, 2016.

[12] T. Chen and C.-W. Shu. Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. Journal of Computational Physics, 345:427-461, 2017.

[13] B. Cockburn, S. Hou and C.-W. Shu. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math-ematics of Computation, 54:545-581, 1990.

[14] B. Cockburn, S.-Y. Lin and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. Journal of Computational Physics, 84:90-113, 1989.

[15] B. Cockburn and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Mathematics of computation, 52:411-435, 1989.

[16] B. Cockburn and C.-W. Shu. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numerical Analysis, 35:2440-2463, 1998.

[17] B. Cockburn and C.-W. Shu. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. Journal of Computational Physics, 141:199-224, 1998.

[18] M. G. Crandall and A. Majda. Monotone difference approximations for scalar conservation laws. Mathematics of Computation, 34:1-21, 1980.

[19] J. Crean, J. E. Hicken, D. C. Del Rey Fernández, D. W. Zingg and M. H. Carpenter. High-order, entropy-stable discretizations of the Euler equations for complex geometries. In 23rd AIAA Computational Fluid Dynamics Conference. American Institute of Aeronautics and Astro-nautics, 2017.

[20] J. Crean, J. E. Hicken, D. C. Del Rey Fernández, D. W. Zingg and M. H. Carpenter. Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements. Journal of Computational Physics, 356:410-438, 2018.

[21] J. Crean, K. Panda, A. Ashley and J. E. Hicken. Investigation of stabilization methods for multi-dimensional summation-by-parts discretizations of the Euler equations. In 54th AIAA Aerospace Sciences Meeting, 2016.

[22] C. M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics, volume 325, Springer, 2010.

[23] D. Derigs, A. R. Winters, G. J. Gassner, S. Walch and M. W. Bohm. Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohy-drodynamics equations. Journal of Computational Physics, 364:420-467, 2018.

[24] D. C. Del Rey Fernández, P. D. Boom and D. W. Zingg. A generalized framework for nodal first derivative summation-by-parts operators. Journal of Computational Physics, 266:214-239, 2014.

[25] D. C. Del Rey Fernández, J. Crean, M. H. Carpenter and J. E Hicken. Staggered-grid entropy-stable multidimensional summation-by-parts discretizations on curvilinear coordi-nates. Journal of Computational Physics, 392:161-186, 2019.

[26] D. C. Del Rey Fernández, J. E. Hicken and D. W Zingg. Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Computers & Fluids, 95:171-196, 2014.

[27] D. C. Del Rey Fernández, J. E. Hicken and D. W Zingg. Simultaneous approximation terms for multi-dimensional summation-by-parts operators. Journal of Scientific Computing, 75:83-110, 2018.

[28] T. C. Fisher. High-order L2 stable multi-domain finite difference method for compressible flows, Ph.D. Thesis, Purdue University, 2012.

[29] T. C. Fisher and M. H. Carpenter. High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains. Journal of Computational Physics, 252:518-557, 2013.

[30] U. S. Fjordholm, S. Mishra and E. Tadmor. Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. Journal of Computational Physics, 230:5587-5609, 2011.

[31] U. S. Fjordholm, S. Mishra and E. Tadmor. Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM Journal on Nu-merical Analysis, 50:544-573, 2012.

[32] U. S. Fjordholm, S. Mishra and E. Tadmor. ENO reconstruction and ENO interpolation are stable. Foundations of Computational Mathematics, 13:139-159, 2013.

[33] U. S. Fjordholm, S. Mishra and E. Tadmor. On the computation of measure-valued solutions. Acta Numerica, 25:567-679, 2016.

[34] L. Friedrich, G. Schnü cke, A. R. Winters, D. C. Del Rey Fernández, G. J. Gassner and M. H. Carpenter. Entropy stable space-time discontinuous Galerkin schemes with summation-by-parts property for hyperbolic conservation laws. Journal of Scientific Com-puting, 80:175-222, 2018.

[35] L. Friedrich, A. R. Winters, D. C. Del Rey Fernández, G. J. Gassner, M. Parsani and M. H Car-penter. An entropy stable h/p non-conforming discontinuous Galerkin method with the summation-by-parts property. Journal of Scientific Computing, 77:689-725, 2018.

[36] G. J. Gassner. A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM Journal on Scientific Computing, 35:A1233-A1253, 2013.

[37] G. J. Gassner, A. R. Winters, F. J. Hindenlang, and D. A. Kopriva. The BR1 scheme is stable for the compressible Navier-Stokes equations. Journal of Scientific Computing, 77:154-200, 2018.

[38] G. J. Gassner, A. R. Winters and D. A Kopriva. Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. Journal of Computational Physics, 327:39-66, 2016.

[39] G. J. Gassner, A. R. Winters and D. A. Kopriva. A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations. Applied Mathematics and Computation, 272:291-308, 2016.

[40] C. Geuzaine and J.-F. Remacle. Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities. International Journal for Numerical Methods in Engineering, 79:1309-1331, 2009.

[41] E. Godlewski and P.-A. Raviart. Hyperbolic Systems of Conservation Laws, Ellipses, 1991.

[42] E. Godlewski and P.-A. Raviart. Numerical Approximation of Hyperbolic Systems of Conservation Laws, volume 118, Springer, 2013.

[43] S. Gottlieb, C.-W. Shu and E. Tadmor. Strong stability-preserving high-order time discretiza-tion methods. SIAM Review, 43:89-112, 2001.

[44] B. Gustafsson, H.-O. Kreiss, and J. Oliger. Time Dependent Problems and Difference Methods, volume 24, John Wiley & Sons, 1995.

[45] A. Harten, B. Engquist, S. Osher and S. R. Chakravarthy. Uniformly high order accurate essentially non-oscillatory schemes, III. In Upwind and High-resolution Schemes, Springer, 1987, pp. 218-290.

[46] A. Harten, J. M. Hyman, P. D. Lax and B. Keyfitz. On finite-difference approximations and entropy conditions for shocks. Communications on Pure and Applied Mathematics, 29:297-322, 1976.

[47] A. Harten, P. D. Lax and B. Van Leer. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review, 25:35-61, 1983.

[48] J. S. Hesthaven, S. Gottlieb and D. Gottlieb. Spectral Methods for Time-dependent Problems, volume 21, Cambridge University Press, 2007.

[49] J. S. Hesthaven and T. Warburton. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer, 2007.

[50] J. E. Hicken. Entropy-stable, high-order discretizations using continuous summation-by-parts operators. In AIAA Aviation 2019 Forum, 2019.

[51] J. E. Hicken and J. Crean. A family of entropy-conservative flux functions for the Euler equations. arXiv preprint, arXiv:1807.03832, 2018.

[52] J. E. Hicken, D. C. Del Rey Fernández and D. W. Zingg. Multidimensional summation-by-parts operators: general theory and application to simplex elements. SIAM Journal on Scientific Computing, 38:A1935-A 1958, 2016.

[53] A. Hiltebrand and S. Mishra. Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws. Numerische Mathematik, 126:103-151, 2014.

[54] S. Hou and X.-D. Liu. Solutions of multi-dimensional hyperbolic systems of conservation laws by square entropy condition satisfying discontinuous Galerkin method. Journal of Scientific Computing, 31:127-151, 2007.

[55] J. Huang and C.-W. Shu. Error estimates to smooth solutions of semi-discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws. Numerical Methods for Partial Differential Equations, 33:467-488, 2017.

[56] T. J. R. Hughes, L. P. Franca, and M. Mallet. A new finite element formulation for compu-tational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. Computer Methods in Applied Mechanics and Engineering, 54:223-234, 1986.

[57] F. Ismail and P. L. Roe. Affordable, entropy-consistent Euler flux functions II: Entropy pro-duction at shocks. Journal of Computational Physics, 228:5410-5436, 2009.

[58] G.-S. Jiang and C.-W. Shu. On a cell entropy inequality for discontinuous Galerkin methods. Mathematics of Computation, 62:531-538, 1994.

[59] D. A. Kopriva. Metric identities and the discontinuous spectral element method on curvi-linear meshes. Journal of Scientific Computing, 26:301-327, 2006.

[60] D. A. Kopriva and G. J. Gassner. On the quadrature and weak form choices in collocation type discontinuous Galerkin spectral element methods. Journal of Scientific Computing, 44:136-155, 2010.

[61] P. G. Lefloch, J.-M. Mercier and C. Rohde. Fully discrete, entropy conservative schemes of arbitrary order. SIAM Journal on Numerical Analysis, 40:1968-1992, 2002.

[62] Y. Liu, C.-W. Shu and M. Zhang. Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes. Journal of Computational Physics, 354:163-178, 2018.

[63] C. Lozano. Entropy production by explicit Runge-Kutta schemes. Journal of Scientific Com-puting, 76:521-564, 2018.

[64] P. Öffner, J. Glaubitz and H. Ranocha. Stability of correction procedure via reconstruc-tion with summation-by-parts operators for Burgers’ equation using a polynomial chaos approach. ESAIM: M2AN, 52:2215-2245, 2018.

[65] S. Ortleb. A kinetic energy preserving DG scheme based on Gauss-Legendre points. Journal of Scientific Computing, 71:1135-1168, 2017.

[66] S. Osher Riemann solvers, the entropy condition, and difference. SIAM Journal on Numerical Analysis, 21:217-235, 1984.

[67] M. Parsani, M. H. Carpenter, T. C. Fisher and E. J. Nielsen. Entropy stable staggered grid discontinuous spectral collocation methods of any order for the compressible Navier-Stokes equations. SIAM Journal on Scientific Computing, 38:A3129-A3162, 2016.

[68] J. Qiu and C.-W. Shu. Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM Journal on Scientific Computing, 26:907-929, 2005.

[69] H. Ranocha. Shallow water equations: split-form, entropy stable, well-balanced, and posi-tivity preserving numerical methods. GEM-International Journal on Geomathematics, 8:85-133, 2017.

[70] H. Ranocha. Generalised summation-by-parts operators and entropy stability of numerical methods for hyperbolic balance laws, Ph.D. Thesis, Technische Universitt Braunschweig, 2018.

[71] H. Ranocha, P. Öffner and T. Sonar. Extended skew-symmetric form for summation-by-parts operators and varying Jacobians. Journal of Computational Physics, 342:13-28, 2017.

[72] H. Ranocha, M. Sayyari, L. Dalcin, M. Parsani and D. I. Ketcheson. Relaxation Runge-Kutta methods: fully-discrete explicit entropy-stable schemes for the Euler and Navier-Stokes equations. SIAM Journal on Scientific Computing, 42:A612-A638, 2020.

[73] F. Renac. Entropy stable DGSEM for nonlinear hyperbolic systems in nonconservative form with application to two-phase flows. Journal of Computational Physics, 382:1-26, 2019.

[74] G. Schnü cke, N. Krais, T. Bolemann and G. J. Gassner. Entropy stable discontinuous Galerkin schemes on moving meshes for hyperbolic conservation laws. Journal of Scientific Computing, 82:69, 2020.

[75] C.-W. Shu. TVB uniformly high-order schemes for conservation laws. Mathematics of Com-putation, 49:105-121, 1987.

[76] C.-W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics, 77:439-471, 1988.

[77] Z. Sun, J. A. Carillo and C.-W. Shu. A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials. Journal of Computational Physics, 352:76-104, 2018.

[78] Z. Sun, J. A. Carillo and C.-W. Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic and Related Models, 12:885-908, 2019.

[79] Z. Sun and C.-W. Shu. Stability analysis and error estimates of Lax-Wendroff discontinuous Galerkin methods for linear conservation laws. ESAIM: Mathematical Modelling and Numeri-cal Analysis, 51:1063-1087, 2017.

[80] M. Svärd and J. Review of summation-by-parts schemes for initial-boundary-value problems. Journal of Scientific Computing, 58:61-89, 2014.

[81] E. Tadmor. The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Mathematics of Computation, 49:91-103, 1987.

[82] E. Tadmor. Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numerica, 12:451-512, 2003.

[83] N. Wintermeyer, A. R. Winters, G. J. Gassner and D. A. Kopriva. An entropy stable nodal dis-continuous Galerkin method for the two dimensional shallow water equations on unstruc-tured curvilinear meshes with discontinuous bathymetry. Journal of Computational Physics, 340:200-242, 2017.

[84] N. Wintermeyer, A. R. Winters, G. J. Gassner and T. Warburton. An entropy stable discontin-uous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs. Journal of Computational Physics, 375:447-480, 2018.

[85] A. R. Winters and G. J. Gassner. A comparison of two entropy stable discontinuous Galerkin spectral element approximations for the shallow water equations with non-constant topog-raphy. Journal of Computational Physics, 301:357-376, 2015.

[86] D. Xiu. Numerical Methods for Stochastic Computations: a Spectral Method Approach, Princeton University Press, 2010.

[87] M. Zakerzadeh andG. May. Entropy stable discontinuous Galerkin scheme for the com-pressible Navier-Stokes equations. In 55th AIAA Aerospace Sciences Meeting, 2017, pp.0084.

[88] L. Zhang, T. Cui and H. Liu. A set of symmetric quadrature rules on triangles and tetrahedra. Journal of Computational Mathematics, 27:89-96, 2009.

[89] Q. Zhang and C.-W. Shu. Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM Journal on Numerical Analysis, 42:641-666, 2004.

[90] Q. Zhang and C.-W. Shu. Error estimates to smooth solutions of Runge-Kutta discontin-uous Galerkin method for symmetrizable systems of conservation laws. SIAM Journal on Numerical Analysis, 44:1703-1720, 2006.

[91] Q. Zhang and C.-W. Shu. Stability analysis and a priori error estimates of the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws. SIAM Journal on Numerical Analysis, 48:1038-1063, 2010.

[92] X. Zhang and C.-W. Shu. On maximum-principle-satisfying high order schemes for scalar conservation laws. Journal of Computational Physics, 229:3091-3120, 2010.

[93] X. Zhang and C.-W. Shu. On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. Journal of Computational Physics, 229:8918-8934, 2010.