On discretely entropy conservative and entropy stable discontinuous Galerkin methods

Abstract 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 [1] , [2] . These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matrices [3] , [4] , [5] , [6] . 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.

[1]  Matteo Parsani,et al.  Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations , 2014, J. Comput. Phys..

[2]  Bo Dong,et al.  Optimal Convergence of the Original DG Method for the Transport-Reaction Equation on Special Meshes , 2008, SIAM J. Numer. Anal..

[3]  Matteo Parsani,et al.  Entropy Stable Staggered Grid Discontinuous Spectral Collocation Methods of any Order for the Compressible Navier-Stokes Equations , 2016, SIAM J. Sci. Comput..

[4]  Sigrun Ortleb.  A Kinetic Energy Preserving DG Scheme Based on Gauss–Legendre Points , 2017, J. Sci. Comput..

[5]  A. Ern,et al.  Mathematical Aspects of Discontinuous Galerkin Methods , 2011 .

[6]  Jesse Chan,et al.  Weight‐adjusted discontinuous Galerkin methods: Matrix‐valued weights and elastic wave propagation in heterogeneous media , 2017, 1701.00215.

[7]  Sigrun Ortleb.  Kinetic energy preserving DG schemes based on summation-by-parts operators on interior node distributions: DGKEP on interior nodes , 2016 .

[8]  Travis C. Fisher,et al.  High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains , 2013, J. Comput. Phys..

[9]  Philipp Öffner,et al.  Extended skew-symmetric form for summation-by-parts operators and varying Jacobians , 2017, J. Comput. Phys..

[10]  Chi-Wang Shu,et al.  On local conservation of numerical methods for conservation laws , 2017, Computers & Fluids.

[11]  A. Harten.  On the symmetric form of systems of conservation laws with entropy , 1983 .

[12]  Timothy C. Warburton,et al.  Weight-Adjusted Discontinuous Galerkin Methods: Wave Propagation in Heterogeneous Media , 2016, SIAM J. Sci. Comput..

[13]  Gregor Gassner,et al.  The BR1 Scheme is Stable for the Compressible Navier–Stokes Equations , 2017, J. Sci. Comput..

[14]  Steven H. Frankel,et al.  Entropy Stable Spectral Collocation Schemes for the Navier-Stokes Equations: Discontinuous Interfaces , 2014, SIAM J. Sci. Comput..

[15]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[16]  Rémi Abgrall,et al.  High‐order CFD methods: current status and perspective , 2013 .

[17]  Axel Modave,et al.  GPU-accelerated discontinuous Galerkin methods on hybrid meshes , 2015, J. Comput. Phys..

[18]  Gregor J. Gassner,et al.  A kinetic energy preserving nodal discontinuous Galerkin spectral element method , 2014 .

[19]  Wim A. Mulder,et al.  A comparison of continuous mass‐lumped finite elements with finite differences for 3‐D wave propagation , 2014 .

[20]  Jesse Chan,et al.  Orthogonal Bases for Vertex-Mapped Pyramids , 2016, SIAM J. Sci. Comput..

[21]  Magnus Svärd,et al.  Entropy-Stable Schemes for the Euler Equations with Far-Field and Wall Boundary Conditions , 2014, J. Sci. Comput..

[22]  David C. Del Rey Fernández,et al.  High-Order , Entropy-Stable Discretizations of the Euler Equations for Complex Geometries , 2017 .

[23]  George Em Karniadakis,et al.  De-aliasing on non-uniform grids: algorithms and applications , 2003 .

[24]  J. Hesthaven,et al.  High–order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem , 2004, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[25]  Philip L. Roe,et al.  Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks , 2009, J. Comput. Phys..

[26]  David C. Del Rey Fernández,et al.  A generalized framework for nodal first derivative summation-by-parts operators , 2014, J. Comput. Phys..

[27]  Timothy C. Warburton,et al.  Weight-Adjusted Discontinuous Galerkin Methods: Curvilinear Meshes , 2016, SIAM J. Sci. Comput..

[28]  Richard Liska,et al.  Comparison of Several Difference Schemes on 1D and 2D Test Problems for the Euler Equations , 2003, SIAM J. Sci. Comput..

[29]  Gregor Gassner,et al.  A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations , 2016, Appl. Math. Comput..

[30]  Zydrunas Gimbutas,et al.  A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions , 2010, Comput. Math. Appl..

[31]  David C. Del Rey Fernández,et al.  Multidimensional Summation-by-Parts Operators: General Theory and Application to Simplex Elements , 2015, SIAM J. Sci. Comput..

[32]  Gregor Gassner,et al.  An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry , 2015, J. Comput. Phys..

[33]  Chi-Wang Shu,et al.  High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems , 2009, SIAM Rev..

[34]  Xu-Dong Liu,et al.  Solution of Two-Dimensional Riemann Problems of Gas Dynamics by Positive Schemes , 1998, SIAM J. Sci. Comput..

[35]  Xiangxiong Zhang,et al.  On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes , 2010, J. Comput. Phys..

[36]  W. A. Mulder,et al.  Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation , 1999 .

[37]  Gregor Gassner,et al.  A Skew-Symmetric Discontinuous Galerkin Spectral Element Discretization and Its Relation to SBP-SAT Finite Difference Methods , 2013, SIAM J. Sci. Comput..

[38]  Timothy C. Warburton,et al.  Nodal discontinuous Galerkin methods on graphics processors , 2009, J. Comput. Phys..

[39]  Gregor Gassner,et al.  Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations , 2016, J. Comput. Phys..

[40]  David C. Del Rey Fernández,et al.  Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements , 2018, J. Comput. Phys..

[41]  Xiangxiong Zhang,et al.  Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes , 2011, Journal of Scientific Computing.

[42]  Eitan Tadmor,et al.  Arbitrarily High-order Accurate Entropy Stable Essentially Nonoscillatory Schemes for Systems of Conservation Laws , 2012, SIAM J. Numer. Anal..

[43]  Deep Ray,et al.  Entropy Stable Scheme on Two-Dimensional Unstructured Grids for Euler Equations , 2016 .

[44]  Chi-Wang Shu,et al.  Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws , 2017, J. Comput. Phys..

[45]  J. Peraire,et al.  Sub-Cell Shock Capturing for Discontinuous Galerkin Methods , 2006 .

[46]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[47]  M. Mock,et al.  Systems of conservation laws of mixed type , 1980 .

[48]  J. Hesthaven,et al.  On the constants in hp-finite element trace inverse inequalities , 2003 .

[49]  Hendrik Ranocha,et al.  Generalised summation-by-parts operators and variable coefficients , 2017, J. Comput. Phys..

[50]  Philipp Öffner,et al.  Summation-by-parts operators for correction procedure via reconstruction , 2015, J. Comput. Phys..

[51]  Mark Ainsworth,et al.  Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods , 2004 .

[52]  Gregor Gassner,et al.  A uniquely defined entropy stable matrix dissipation operator for high Mach number ideal MHD and compressible Euler simulations , 2017, J. Comput. Phys..

[53]  Eitan Tadmor,et al.  The numerical viscosity of entropy stable schemes for systems of conservation laws. I , 1987 .

[54]  Lilia Krivodonova,et al.  Limiters for high-order discontinuous Galerkin methods , 2007, J. Comput. Phys..

[55]  Georg Stadler,et al.  A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media , 2010, J. Comput. Phys..

[56]  Hendrik Ranocha,et al.  Comparison of Some Entropy Conservative Numerical Fluxes for the Euler Equations , 2017, J. Sci. Comput..

[57]  M. Carpenter,et al.  Fourth-order 2N-storage Runge-Kutta schemes , 1994 .

[58]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[59]  Eitan Tadmor,et al.  Solution of two‐dimensional Riemann problems for gas dynamics without Riemann problem solvers , 2002 .

[60]  Wang Chi-Shu,et al.  Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws , 1997 .

[61]  Timothy C. Warburton,et al.  A Low-Storage Curvilinear Discontinuous Galerkin Method for Wave Problems , 2013, SIAM J. Sci. Comput..

[62]  Juhani Pitkäranta,et al.  An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation , 1986 .

[63]  Praveen Chandrashekar,et al.  Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations , 2012, ArXiv.

[64]  T. Hughes,et al.  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 , 1986 .