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Split Form ALE DG Methods for the Euler Equations: Entropy Stability and Kinetic Energy Dissipation

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Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 137))

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Abstract

The construction of discontinuous Galerkin (DG) methods for the compressible Euler equations includes the approximation of non-linear flux terms in the volume integrals. The terms can lead to aliasing and stability issues. The entropy and kinetic energy are elevated in smooth, but under-resolved parts of the solution which are affected by aliasing. In this work the split form DG framework is used to construct entropy conservative (EC) or kinetic energy preserving (KEP) nodal discontinuous Galerkin spectral element methods (DGSEM) to solve the Euler equations on moving hexahedral meshes. The Arbitrary Lagrangian Eulerian (ALE) approach is used to include the effect of mesh motion in the split form DG methods. Since the EC or KEP property is not sufficient to tame discontinuities in the numerical solution, the split form ALE DGSEM are modified by adding numerical dissipation matrices to the EC or KEP surface numerical fluxes. This leads to entropy stable (ES) or kinetic energy dissipative (KED) methods. The three dimensional Taylor-Green vortex (TGV) is investigated to analyze the properties of the constructed split form ALE DGSEM.

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References

  1. Blaisdell, G.A., Spyropoulos, E.T., Qin, J.H.: The effect of the formulation of nonlinear terms on aliasing errors in spectral methods. Appl. Numer. Math. 21(3), 207–219 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods. In: Fundamentals in Single Domains. Springer, Berlin (2006)

    Google Scholar 

  3. Chandrashekar, P.: Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations. Commun. Comput. Phys. 14(5), 1252–1286 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Donea, J., Huerta, A., Ponthot, J.-P., Rodríguez-Ferran, A.: Arbitrary Lagrangian-Eulerian methods. In: Encyclopedia of Computational Mechanics, 2nd edn., pp.1–32 (2017)

    Google Scholar 

  6. Fernández, D.C.D.R., Hicken, J.E., Zingg, D.W.: Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids 95, 171–196 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. J. Comput. Phys. 252, 518–557 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gassner, G.J.: A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM J. Sci. Comput. 35(3), A1233–A1253 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gassner, G.J., Winters, A.R., Kopriva, D.A.: Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. J. Comput. Phys. 327, 39–66 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gassner, G.J., Winters, A.R., Hindenlang, F.J., Kopriva, D.A.: The BR1 scheme is stable for the compressible Navier-Stokes equations. J. Sci. Comput. 77(1), 154–200 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Harten, A.: On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49, 151–164 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2007)

    MATH  Google Scholar 

  13. Jameson, A.: 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. J. Sci. Comput. 34(2), 188–208 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kennedy, C.A., Gruber, A.: Reduced aliasing formulations of the convective terms within the Navier–Stokes equations for a compressible fluid. J. Comput. Phys. 227(3), 1676–1700 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kennedy, C.A., Carpenter, M.H., Lewis, R.M.: Low-storage, explicit Runge-Kutta schemes for the compressible Navier-Stokes equations. Appl. Numer. Math. 35(3), 177–219 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kirby, R.M., Karniadakis, G.E.: De-aliasing on non-uniform grids: algorithms and applications. J. Comput. Phys. 191(1), 249–264 (2003)

    Article  MATH  Google Scholar 

  17. Kopriva, D.A.: Metric identities and the discontinuous spectral element method on curvilinear meshes. J. Sci. Comput. 26(3), 301 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  19. Kopriva, D.A., Winters, A.R., Bohm, M., Gassner, G.J.: A provably stable discontinuous Galerkin spectral element approximation for moving hexahedral meshes. Comput. Fluids 139, 148–160 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Krais, N., Beck, A., Bolemann, T., Frank, H., Flad, D., Gassner, G., Hindenlang, F., Hoffmann, M., Kuhn, T., Sonntag, M., Munz, C.-D.: FLEXI: a high order discontinuous Galerkin framework for hyperbolic-parabolic conservation laws (2019). ar**v:1910.02858

    Google Scholar 

  21. Krais, N., Schnücke, G., Bolemann, T., Gassner, G.J.: Split form ALE discontinuous Galerkin methods with applications to under-resolved turbulent low-Mach number flows. J. Comput. Phys. 421, 109726 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kreiss, H.-O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24(3), 199–215 (1972)

    Article  MathSciNet  Google Scholar 

  23. Lefloch, P.G., Mercier, J.-M., Rohde, C.: Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40(5), 1968–1992 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16–42 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lomtev, I., Kirby, R.M., Karniadakis, G.E.: A discontinuous Galerkin ALE method for compressible viscous flows in moving domains. J. Comput. Phys. 155(1), 128–159 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Merriam, M.L.: Towards a rigorous approach to artificial dissipation. National Aeronautics and Space Administration, Moffett Field. Ames Research Center (1989)

    Google Scholar 

  27. Minoli, C.A.A., Kopriva, D.A.: Discontinuous Galerkin spectral element approximations on moving meshes. J. Comput. Phys. 230(5), 1876–1902 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nguyen, V.-T.: An arbitrary Lagrangian–Eulerian discontinuous Galerkin method for simulations of flows over variable geometries. J. Fluids Struct. 26(2), 312–329 (2010)

    Article  Google Scholar 

  29. Nordström, J., Carpenter, M.H.: Boundary and interface conditions for high-order finite-difference methods applied to the Euler and Navier-Stokes equations. J. Comput. Phys. 148(2), 621–645 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Persson, P.-O., Bonet, J., Peraire, J.: Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains. Comput. Methods Appl. Mech. Eng. 198(17–20), 1585–1595 (2009)

    Article  MATH  Google Scholar 

  31. Pirozzoli, S.: Generalized conservative approximations of split convective derivative operators. J. Comput. Phys. 229(19), 7180–7190 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ranocha, H.: Generalised summation-by-parts operators and entropy stability of numerical methods for hyperbolic balance laws. Cuvillier Verlag, Göttingen (2018)

    Google Scholar 

  33. Schnücke, G., Krais, N., Bolemann, T., Gassner, G.J.: Entropy stable discontinuous Galerkin schemes on moving meshes for hyperbolic conservation laws. J. Sci. Comput. 82, 1–42 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shu, C.-W., Don, W.-S., Gottlieb, D., Schilling, O., Jameson, L.: Numerical convergence study of nearly incompressible, inviscid Taylor-Green vortex flow. J. Sci. Comput. 24(1), 1–27 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tang, T.: Moving mesh methods for computational fluid dynamics. Contemp. Math. 383(8), 141–173 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research is supported by the European Research Council (ERC) under the European Union’s Eights Framework Program Horizon 2020 with the research project Extreme, ERC grant agreement no. 714487. The authors gratefully acknowledge the support and the computing time on “Hazel Hen” provided by the High-Performance Computing Center Stuttgart (HLRS) through the project hpcdg”.

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Jena, Jena, Germany

    Gero Schnücke

  2. Department for Mathematics/Computer Science, Center for Data and Simulation Science, University of Cologne, Cologne, Germany

    Gregor J. Gassner

  3. Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart, Stuttgart, Germany

    Nico Krais

Authors
  1. Gero Schnücke
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  2. Gregor J. Gassner
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  3. Nico Krais
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Correspondence to Gero Schnücke .

Editor information

Editors and Affiliations

  1. TU Wien, Wien, Austria

    Jens M. Melenk

  2. Universität Wien, Wien, Austria

    Ilaria Perugia

  3. TU Wien, Wien, Austria

    Joachim Schöberl

  4. ETH Zürich, Zürich, Switzerland

    Christoph Schwab

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Schnücke, G., Gassner, G.J., Krais, N. (2023). Split Form ALE DG Methods for the Euler Equations: Entropy Stability and Kinetic Energy Dissipation. In: Melenk, J.M., Perugia, I., Schöberl, J., Schwab, C. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1. Lecture Notes in Computational Science and Engineering, vol 137. Springer, Cham. https://doi.org/10.1007/978-3-031-20432-6_27

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