An Entropy Stable h / p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property

This work presents an entropy stable discontinuous Galerkin (DG) spectral element approximation for systems of non-linear conservation laws with general geometric (h) and polynomial order (p) non-conforming rectangular meshes. The crux of the proofs presented is that the nodal DG method is constructed with the collocated Legendre–Gauss–Lobatto nodes. This choice ensures that the derivative/mass matrix pair is a summation-by-parts (SBP) operator such that entropy stability proofs from the continuous analysis are discretely mimicked. Special attention is given to the coupling between non-conforming elements as we demonstrate that the standard mortar approach for DG methods does not guarantee entropy stability for non-linear problems, which can lead to instabilities. As such, we describe a precise procedure and modify the mortar method to guarantee entropy stability for general non-linear hyperbolic systems on h / p non-conforming meshes. We verify the high-order accuracy and the entropy conservation/stability of fully non-conforming approximation with numerical examples.

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  Claus-Dieter Munz,et al.  Explicit Discontinuous Galerkin methods for unsteady problems , 2012 .

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

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

[5]  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..

[6]  Spencer J. Sherwin,et al.  A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations , 2017, J. Comput. Phys..

[7]  David C. Del Rey Fernández,et al.  Conservative and Stable Degree Preserving SBP Finite Difference Operators for Non-Conforming Meshes , 2016 .

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

[9]  Eitan Tadmor,et al.  Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography , 2011, J. Comput. Phys..

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

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

[12]  Gregor Gassner,et al.  Conservative and Stable Degree Preserving SBP Operators for Non-conforming Meshes , 2016, Journal of Scientific Computing.

[13]  Nail K. Yamaleev,et al.  Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: Theory and boundary conditions , 2013, J. Comput. Phys..

[14]  Matteo Parsani,et al.  Towards an Entropy Stable Spectral Element Framework for Computational Fluid Dynamics , 2016 .

[15]  David A. Kopriva,et al.  Implementing Spectral Methods for Partial Differential Equations , 2009 .

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

[17]  Gregor Gassner,et al.  An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: Theory and numerical verification , 2018, J. Comput. Phys..

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

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

[20]  David Gottlieb,et al.  Spectral Methods on Arbitrary Grids , 1995 .

[21]  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..

[22]  E. Tadmor.  Skew-selfadjoint form for systems of conservation laws , 1984 .

[23]  Mark H. Carpenter,et al.  Stable and Accurate Interpolation Operators for High-Order Multiblock Finite Difference Methods , 2009, SIAM J. Sci. Comput..

[24]  Jeremy E. Kozdon,et al.  Provably stable, general purpose projection operators for high-order finite difference methods , 2015 .

[25]  Davis A. Kopriva,et al.  Computation of electromagnetic scattering with a non‐conforming discontinuous spectral element method , 2002 .

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

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

[28]  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..

[29]  Omar Ghattas,et al.  Analysis of an hp-Nonconforming Discontinuous Galerkin Spectral Element Method for Wave Propagation , 2012, SIAM J. Numer. Anal..

[30]  Matteo Parsani,et al.  Entropy stable discontinuous interfaces coupling for the three-dimensional compressible Navier-Stokes equations , 2015, J. Comput. Phys..

[31]  Jan Nordström,et al.  On the Suboptimal Accuracy of Summation-by-parts Schemes with Non-conforming Block Interfaces , 2016 .

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

[33]  John H. Kolias,et al.  A CONSERVATIVE STAGGERED-GRID CHEBYSHEV MULTIDOMAIN METHOD FOR COMPRESSIBLE FLOWS , 1995 .

[34]  H. C. Yee.  Skew-Symmetric Splitting and Stability of High Order Central Schemes , 2017 .

[35]  Tadmor.  Entropy functions for symmetric systems of conservation laws. Final Report , 1987 .

[36]  Deep Ray,et al.  An entropy stable finite volume scheme for the two dimensional Navier-Stokes equations on triangular grids , 2017, Appl. Math. Comput..