Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations

Fisher and Carpenter (2013) [12] found a remarkable equivalence of general diagonal norm high-order summation-by-parts operators to a subcell based high-order finite volume formulation. This equivalence enables the construction of provably entropy stable schemes by a specific choice of the subcell finite volume flux. We show that besides the construction of entropy stable high-order schemes, a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms. Thus, by changing the subcell finite volume flux to a specific choice, we are able to generate, in a systematic way, all common split forms of the compressible Euler advection terms, such as the Ducros splitting and the Kennedy and Gruber splitting. Although these split forms are not entropy stable, we present a systematic way to prove which of those split forms are at least kinetic energy preserving. With this, we construct a unified high-order split form DG framework. We investigate with three dimensional numerical simulations of the inviscid TaylorGreen vortex and show that the new split forms enhance the robustness of high-order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows. In fact, we show that for certain test cases, the novel split form discontinuous Galerkin schemes are more robust than the discontinuous Galerkin scheme with over-integration.

[1]  Travis Calob Fisher,et al.  High-order L2 stable multi-domain finite difference method for compressible flows , 2012 .

[2]  Christopher A. Kennedy,et al.  Reduced aliasing formulations of the convective terms within the Navier-Stokes equations for a compressible fluid , 2008, J. Comput. Phys..

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

[4]  Sergio Pirozzoli,et al.  Generalized conservative approximations of split convective derivative operators , 2010, J. Comput. Phys..

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

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

[7]  David A. Kopriva,et al.  Metric Identities and the Discontinuous Spectral Element Method on Curvilinear Meshes , 2006, J. Sci. Comput..

[8]  Yohei Morinishi,et al.  Skew-symmetric form of convective terms and fully conservative finite difference schemes for variable density low-Mach number flows , 2010, J. Comput. Phys..

[9]  Gregor Gassner,et al.  An Energy Stable Discontinuous Galerkin Spectral Element Discretization for Variable Coefficient Advection Problems , 2014, SIAM J. Sci. Comput..

[10]  Antony Jameson,et al.  Formulation of Kinetic Energy Preserving Conservative Schemes for Gas Dynamics and Direct Numerical Simulation of One-Dimensional Viscous Compressible Flow in a Shock Tube Using Entropy and Kinetic Energy Preserving Schemes , 2008, J. Sci. Comput..

[11]  Vincent Guinot,et al.  High-Order Fluxes for Conservative Skew-Symmetric-like Schemes in Structured Meshes , 2000 .

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

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

[14]  Chi-Wang Shu,et al.  On a cell entropy inequality for discontinuous Galerkin methods , 1994 .

[15]  Gregor Gassner,et al.  On the Quadrature and Weak Form Choices in Collocation Type Discontinuous Galerkin Spectral Element Methods , 2010, J. Sci. Comput..

[16]  P. Moin,et al.  Effect of numerical dissipation on the predicted spectra for compressible turbulence , 2022 .

[17]  T. A. Zang,et al.  On the rotation and skew-symmetric forms for incompressible flow simulations , 1991 .

[18]  Siddhartha Mishra,et al.  Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws , 2014, Numerische Mathematik.

[19]  Gregory A. Blaisdell,et al.  The effect of the formulation of nonlinear terms on aliasing errors in spectral methods , 1996 .

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

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

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

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

[24]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[25]  Spencer J. Sherwin,et al.  Dealiasing techniques for high-order spectral element methods on regular and irregular grids , 2015, J. Comput. Phys..

[26]  P. Moin,et al.  On the Effect of Numerical Errors in Large Eddy Simulations of Turbulent Flows , 1997 .

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

[28]  David A. Kopriva,et al.  Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers , 2009 .

[29]  B. Strand.  Summation by parts for finite difference approximations for d/dx , 1994 .

[30]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[31]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[32]  A. Beck,et al.  On the accuracy of high-order discretizations for underresolved turbulence simulations , 2013 .

[33]  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..

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

[35]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[36]  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..

[37]  Chi-Wang Shu,et al.  The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws , 1988, ESAIM: Mathematical Modelling and Numerical Analysis.

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

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