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## Speaker

Helen C. Yee, Senior Research Scientist, NASA AMES Research Center

## Title

Recent Advancement of Entropy Split Methods for Compressible Gas Dynamics and MHD

## Abstract

The entropy splitting of the compressible Euler flux derivatives based on Harten’s entropy function [6, 4, 24] in conjunction with classical spatial central and DRP (dispersion relation-preserving) finite discretizations with summation-by-parts (SBP) operators [22] for both periodic and non-periodic boundary conditions is proven to be entropy conservative and stable for a thermally-perfect gas by Sjogreen & Yee [18, 14, 19]. The various high order methods resulting from applying classical spatial central, DRP and Pade (compact) methods to the split form of the Euler flux derivative are referred to as entropy split methods as a function of the splitting parameter β. These entropy split methods are entropy conserving and stable but they are usually not conservative numerical methods without additional reformulation; e.g., those proposed in Sjogreen & Yee [19].

Here the same Harten entropy function is used for the ideal magnetohydrodynamic (MHD) governing equation set. This approach was not examined in great depth by Sjogreen & Yee and Yee & Sj¨ogreen in two conference proceedings papers [21, 27], which also included the high order Pade (compact) spatial dis-cretizations [7]. For the extension of the entropy split method to the ideal MHD, the Godunov symmetriz-able non-conservative form of the formulation is used. Due to the non-conservative portion (commonly referred to as a nonlinear source term vector) of the symmetrizable system [5], there are variants in formu-lating the entropy split methods. Two different numerical treatments for these particular source terms of the symmetrizable MHD governing equation set are presented.

The comparative studies concentrate on the Tadmor-type of discrete entropy conserving [23], momentum conserving [2], kinetic energy preserving [8, 10, 1], Yee et al., Sjogreen & Yee entropy split methods [24, 16, 13, 18, 14, 19], as well as the combination of these physical-preserving methods [11, 26]. All of these methods are not only preserve certain physical properties of the chosen governing equations but are also known to either improve numerical stability, and/or minimize aliasing errors in long time integration of turbulent flow computations without the aid of added numerical dissipation for selected compressible flow types. Extensive error norm comparison with grid refinement were performed to show how well nine of the eighth-order methods conserve the entropy, momentum and mass, and preserve the kinetic energy for long time integration of the various flows. In general the Tadmor-type entropy conserving methods are at least twice the CPU per time step than the rest of the considered nine methods

References

1. Coppola, G., Capuano, F., Pirozzoli, S., de Luca, L.: Numerically Stable Formulations of Convective Terms for Turbulent Compressible Flows. J. Comput. Phys., 2019, DOI:10.1016/j.jcp.2019.01.007.

2. Ducros, F., Laporte, F., Souleres, T., Guinot, V., Moinat, P., Caruelle, B.: High-Order Fluxes for Conservative Skew-Symmetric-Like Schemes in Structured Meshes: Application to Compressible Flows, J. Comput. Phys. 161, 114-139 (2000).

3. Gassner, G., Winters, A.R.: A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics: Why? When? What? Where?, Frontiers in Physics 8:500690, DOI:10.3389/fphy.2020.500690, Jan. 2021.

4. Gerritsen, M, Olsson, P.: Designing an Efficient Solution Strategy for Fluid Flows. I. A Stable High Order Finite Differ-ence Scheme and Sharp Shock Resolution for the Euler Equations. J. Comput. Phys. 129, 245-262 (1996).

5. Godunov, S.K.: Symmetric Form of the Equations of Magnetohydrodynamics. Numerical Methods for Mechanics of Continuum Medium, 13, no. 1, 26-34 (1972).

6. Harten, A: On the Symmetric Form of Systems for Conservation Laws with Entropy: J. Comput Phys. 49, 151 (1983).

7. Hirsh, RS.: Higher Order Accurate Difference Solutions of Fluid Mechanics Problems by a Compact Differencing Eech-nique. J Comput Phys. 19, 90-109 (1975).

8. Kennedy, C.A. and Gruber, A.: Reduced Aliasing Formulations of the Convective Terms Within the Navier-Stokes Equa-tions. J. Comput. Phys., 227 1676-1700 (2008).

9. Olsson, P., Oliger, J.: Energy and maximum norm estimates for nonlinear conservation laws. RIACS Technical Report 94.01, (1994).

10. Pirozzoli, S.: Generalized conservative approximations of split convective derivative operators. J. Comput. Phys., 219, 7180-90 (2010).

11. Ranocha. H.: Entropy Conserving and Kinetic Energy Preserving Numerical Methods for the Euler Equations Using Summation-by-Parts Operators. Proceedings of the ICOSAHOM-2018, Imperial College, London, UK, July 9-13, 2018.

12. Sjogreen, B., Yee, H.C.: An entropy stable method revisited: Central differencing via entropy splitting and SBP. Entropy Splitting for Central Schemes with SBP Energy Estimates & Skew-Symmetric Splitting Revist. Proceeding of Proceeding of ICOSAHOM-2018, July 9-13, 2018, Imperial College, London, UK.

13. Sjogreen, B., Yee, H.C.:High order entropy conservative central schemes for wide ranges of compressible gas dynamics and MHD flows. J. Comput. Phys., 364, 153-185 (2018).

14. Sjogreen, B., Yee, H.C.: Skew-Symmetric Splitting for Multiscale Gas Dynamics and MHD Turbulence Flows. Extended version of Proceedings of ASTRONUM-2016, June 6-10, 2016, Monterey, CA, USA, J. Scientific Computing, 83 43 (2020).

15. Sjogreen, B., Yee, H.C.: Accuracy Consideration by DRP Schemes for DNS and LES of Compressible Flow Computa-tions. Special issue in Computers & Fluids in honor of Prof. Toro’s 70th birthday, 159 123-136 (2017).

16. Sjogreen, B., Yee, H.C.: . High order entropy conserving central schemes for wide ranges of compressible gas dynamics and MHD flows. J. Comput. Phys., 364, 153-185 (2018).

17. Sjogreen, B., Yee, H.C.: Entropy Stable Method for the Euler Equations Revisited: Central Differencing via Entropy Splitting and SBP. Proceeding of ICOSAHOM-2018, July 9-13, 2018, Imperial College, London, UK; J. Scientific Com-puting, doi.org/10.1007/s10915-019-01013-1, 81 1359-1385 (2019).

18. Sjogreen, B., Yee, H.C.: Entropy Stable Method for the Euler Equations Revisited: Central Differencing via Entropy Splitting and SBP. J. Scientific Computing, 81:1359-1385 (2019), https://doi.org/10.1007/s10915-019-01013-1.

19. Sjogreen, B., Yee, H.C.: Construction of Conservative Numerical Fluxes for the Entropy Split Method. Comm. Applied Mathematics and Computation (CAMC), 2021.

20. Sjogreen, B., Yee, H.C., Kotov, D.V., and A. Kritsuk, A. G.: Skew-Symmetric Splitting for Multiscale Gas Dynamics and MHD Turbulence Flows. J. Scientific Computing, 83:1-43, (2020)

21. Sjogreen, B., Yee, H.C.: High Order Compact Central Spatial Discretization Under the Framework of Entropy Split Methods. Proceedings of the ICOSAHOM21, July 12-16, 2021.

22. Strand, B.: Summation by Parts for Finite Difference Approximations for d/dx. J. Comput. Phys. 110, 47–97, (1996).

23. Tadmor, E.: Entropy Stability Theory for Difference Approximations of Nonlinear Conservation Laws and Related Time-Dependent Problems. Acta Numerica 12, 451-512 (2003).

24. Yee, H. C., Vinokur, M, Djomehri, M. J.: Entropy Splitting and Numerical Dissipation. J. Comp. Phys. 162, 33-81 (2000).

25. Yee, H. C., Sjogreen, B.: Development of Low Dissipative High Order Filter Schemes for Multiscale Navier-Stokes and MHD Systems. J. Comput. Phys., 225 910-934 (2007).

26. Yee, H. C., Sjogreen, B.: On Entropy Conservation and Kinetic Energy Preservation Methods. Proceedings of the ICOSAHOM-2019, July 1-5, 2019, Paris, France.

27. Yee, H. C., Sjogreen, B.: Comparative Study on a Variety of Structure-Preserving High Order Spatial Discretizations with the Entropy Split Methods for MHD. Proceedings of the ICOSAHOM21, July 12-16, 2021.