Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes

For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative semidiscretisations by adding suitable correction terms has been proposed recently by Abgrall (J. Comp. Phys. 372: pp. 640-666, 2018). Here, these correction terms are characterised as solutions of certain optimisation problems and adapted to discontinuous element based schemes such as discontinuous Galerkin and (multi-block) finite difference methods. Novel generalisations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed and tested in numerical experiments. Finally, the underlying idea to use optimisation problems is applied to grid refinement and coarsening operators, resulting in entropy stable/dissipative grid transfers.

[1]  Rémi Abgrall,et al.  A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes , 2017, J. Comput. Phys..

[2]  Hendrik Ranocha,et al.  Shallow water equations: split-form, entropy stable, well-balanced, and positivity preserving numerical methods , 2016, GEM - International Journal on Geomathematics.

[3]  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..

[4]  R. Abgrall,et al.  High Order Schemes for Hyperbolic Problems Using Globally Continuous Approximation and Avoiding Mass Matrices , 2017, J. Sci. Comput..

[5]  Rémi Abgrall,et al.  Explicit Runge-Kutta residual distribution schemes for time dependent problems: Second order case , 2010, J. Comput. Phys..

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

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

[8]  David I. Ketcheson,et al.  Highly Efficient Strong Stability-Preserving Runge-Kutta Methods with Low-Storage Implementations , 2008, SIAM J. Sci. Comput..

[9]  Philipp Öffner,et al.  L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document} Stability of Explicit Runge–Kutta Schemes , 2017, Journal of Scientific Computing.

[10]  Gregor Gassner,et al.  An Entropy Stable h / p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property , 2017, J. Sci. Comput..

[11]  Björn Sjögreen,et al.  High order entropy conservative central schemes for wide ranges of compressible gas dynamics and MHD flows , 2018, J. Comput. Phys..

[12]  P. Heinisch,et al.  Numerical Methods for the Magnetic Induction Equation with Hall Effect and Projections onto Divergence-Free Vector Fields , 2018, 1810.01397.

[13]  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..

[14]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

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

[16]  David C. Del Rey Fernández,et al.  Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations , 2014 .

[17]  Jason E. Hicken,et al.  A Family of Entropy-Conservative Flux Functions for the Euler Equations , 2018, 1807.03832.

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

[19]  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..

[20]  Lisandro Dalcin,et al.  Relaxation Runge-Kutta Methods: Fully-Discrete Explicit Entropy-Stable Schemes for the Euler and Navier-Stokes Equations , 2019, 1905.09129.

[21]  Jesse Chan,et al.  On discretely entropy conservative and entropy stable discontinuous Galerkin methods , 2017, J. Comput. Phys..

[22]  Hendrik Ranocha,et al.  On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators , 2018, IMA Journal of Numerical Analysis.

[23]  David I. Ketcheson,et al.  Relaxation Runge-Kutta Methods: Conservation and Stability for Inner-Product Norms , 2019, SIAM J. Numer. Anal..

[24]  N. Ron-Ho,et al.  A Multiple-Grid Scheme for Solving the Euler Equations , 1982 .

[25]  Iain Dunning,et al.  JuMP: A Modeling Language for Mathematical Optimization , 2015, SIAM Rev..

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

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

[28]  H. Freistühler.  C.M. Dafermos: “Hyperbolic Conservation Laws in Continuum Physics” , 2018, Jahresbericht der Deutschen Mathematiker-Vereinigung.

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

[30]  Rémi Abgrall,et al.  High Order Fluctuation Schemes on Triangular Meshes , 2003, J. Sci. Comput..

[31]  Magnus Svärd,et al.  Review of summation-by-parts schemes for initial-boundary-value problems , 2013, J. Comput. Phys..

[32]  Gregor Gassner,et al.  Entropy Stable Space–Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws , 2018, J. Sci. Comput..

[33]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[34]  Lisandro Dalcin,et al.  Relaxation Runge-Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations , 2019, SIAM J. Sci. Comput..

[35]  Sergio Pirozzoli,et al.  Numerical Methods for High-Speed Flows , 2011 .

[36]  Remi Abgrall,et al.  A high-order nonconservative approach for hyperbolic equations in fluid dynamics , 2017, Computers & Fluids.

[37]  Rémi Abgrall,et al.  Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes , 2011, J. Comput. Phys..