Abstract
The entropy split method is based on the physical entropies of the thermally perfect gas Euler equations. The Euler flux derivatives are approximated as a sum of a conservative portion and a non-conservative portion in conjunction with summation-by-parts (SBP) difference boundary closure of (Gerritsen and Olsson in J Comput Phys 129: 245–262, 1996; Olsson and Oliger in RIACS Tech Rep 94.01, 1994; Yee et al. in J Comp Phys 162: 33–81, 2000). Sjögreen and Yee (J Sci Comput https://doi.org/10.1007/s10915-019-01013-1) recently proved that the entropy split method is entropy conservative and stable. Standard high-order spatial central differencing as well as high order central spatial dispersion relation preserving (DRP) spatial differencing is part of the entropy stable split methodology framework. The current work is our first attempt to derive a high order conservative numerical flux for the non-conservative portion of the entropy splitting of the Euler flux derivatives. Due to the construction, this conservative numerical flux requires higher operations count and is less stable than the original semi-conservative split method. However, the Tadmor entropy conservative (EC) method (Tadmor in Acta Numerica 12: 451–512, 2003) of the same order requires more operations count than the new construction. Since the entropy split method is a semi-conservative skew-symmetric splitting of the Euler flux derivative, a modified nonlinear filter approach of (Yee et al. in J Comput Phys 150: 199–238, 1999, J Comp Phys 162: 3381, 2000; Yee and Sjögreen in J Comput Phys 225: 910934, 2007, High Order Filter Methods for Wide Range of Compressible flow Speeds. Proceedings of the ICOSAHOM09, June 22–26, Trondheim, Norway, 2009) is proposed in conjunction with the entropy split method as the base method for problems containing shock waves. Long-time integration of 2D and 3D test cases is included to show the comparison of these new approaches.
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References
Arakawa, A.: Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comput. Phys. 1, 119–143 (1966)
Balsara, D., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)
Blaisdell, G.A., Spyropoulos, E.T., Qin, J.H.: The effect of the formulation of nonlinear terms on aliasing errors in spectral methods. Appl. Num. Math. 21, 207–219 (1996)
Coppola, G., Capuano, F., Pirozzoli, S., de Luca, L.: Numerically stable formulations of convective terms for turbulent compressible flows. J. Comput. Phys. 382, 86–104 (2019). https://doi.org/10.1016/j.jcp.2019.01.007
Ducros, F., Laporte, F., Soulères, 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)
Gerritsen, M., Olsson, P.: Designing an efficient solution strategy for fluid flows. I. A stable high order finite difference scheme and sharp shock resolution for the Euler equations. J. Comput. Phys. 129, 245–262 (1996)
Harten, A.: On the symmetric form of systems for conservation laws with entropy. J. Comput. Phys. 49, 151–164 (1983)
Kennedy, C.A., Gruber, A.: Reduced aliasing formulations of the convective terms within the Navier-Stokes equations. J. Comput. Phys. 227, 1676–1700 (2008)
Kotov, D.V., Yee, H.C., Wray, A.A., Sjögreen, B., Kritsuk, A.G.: Numerical dissipation control in high order shock-capturing schemes for LES of low speed flows. J. Comput. Phys. 307, 189–202 (2016)
Kotov, D.V., Yee, H.C., Wray, A.A., Sjögreen, B.: High order numerical methods for dynamic SGS model of turbulent flows with shocks. Commun. Comput. Phys. 19, 273–300 (2016)
Olsson, P., Oliger, J.: Energy and maximum norm estimates for nonlinear conservation laws. RIACS Tech. Rep. 94.01 (1994)
Pirozzoli, S.: Generalized conservative approximations of split convective derivative operators. J. Comput. Phys. 219, 7180–90 (2010)
Ranocha, H.: Entropy conserving and kinetic energy preserving numerical methods for the Euler equations using summation-by-parts operators. In: Proceedings of the ICOSAHOM-2018, July 9–13, Imperial College, London, UK (2018)
Sandham, N.D., Li, Q., Yee, H.C.: Entropy splitting for high-order numerical simulation of compressible turbulence. J. Comput. Phys. 23, 307–322 (2002)
Shu, C.-W., Osher, S.J.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 83, 32–78 (1989)
Sjögreen, B., Yee, H.C.: Multiresolution wavelet based adaptive numerical dissipation control for high order methods. J. Sci. Comput. 20, 211–255 (2004)
Sjögreen, B., Yee, H.C., Vinokur, M.: On high order finite-difference metric discretizations satisfying GCL on moving and deforming grids. J. Comput. Phys. 265, 211–220 (2014)
Sjögreen, B., Yee, H.C.: On skew-symmetric splitting and entropy conservation schemes for the Euler equations. In: Proceedings of ENUMATH09, June 29–July 2, Uppsala University, Sweden (2009)
Sjögreen, 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)
Sjögreen, B., Yee, H.C.: Accuracy consideration by DRP schemes for DNS and LES of compressible flow computations. Comput. Fluids 159, 123–136 (2017)
Sjögreen, B., Yee, H.C., Kotov, D., Kritsuk, A.G.: Skew-symmetric splitting for multiscale gas dynamics and MHD turbulence flows. J. Sci. Comput. 83, 43 (2020)
Sjögreen, 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)
Sjögreen, B., Yee, H.C.: An entropy stable method revisited: central differencing via entropy splitting and SBP. In: Proceeding of ICOSAHOM-2018, July 9–13, Imperial College, London UK (2018)
Sjögreen, B., Yee, H.C.: Entropy stable method for the Euler equations revisited: central differencing via entropy splitting and SBP. J. Sci. Comput. 81, 1359–1385 (2019). https://doi.org/10.1007/s10915-019-01013-1
Sandham, N.D., Li, Q., Yee, H.C.: Entropy splitting for high-order numerical simulation of compressible turbulence. J. Comput. Phys. 178, 307–322 (2002)
Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numerica 12, 451–512 (2003)
Taylor, G., Green, A.: Mechanism of the production of small eddies from large ones. Proc. R. Soc. Lond. A 158, 499–521 (1937)
Tauber, E., Sandham, N.D.: Comparison of three large-eddy simulations of shock-induced turbulent separation bubbles. Shock Waves 19, 469–478 (2009)
Yee, H.C., Sandham, N.D., Djomehri, M.J.: Low-dissipative high order shock-capturing methods using characteristic-based filters. J. Comput. Phys. 150, 199–238 (1999)
Yee, H.C., Vinokur, M., Djomehri, M.J.: Entropy splitting and numerical dissipation. J. Comput. Phys. 162, 33–81 (2000)
Yee, H.C., Sjögreen, B.: Development of low dissipative high order filter schemes for multiscale Navier-Stokes and MHD systems. J. Comput. Phys. 225, 910–934 (2007)
Yee, H. C., Sjögreen, B.: High order filter methods for wide range of compressible flow speeds. In: Proceedings of the ICOSAHOM09, June 22–26, Trondheim, Norway (2009)
Yee, H.C., Sjögreen, B.: Recent developments in accuracy and stability improvement of nonlinear filter methods for DNS and LES of compressible flows. Comput. Fluids 169, 331–348 (2018)
Yee, H. C., Sjögreen, B.: On entropy conservation and kinetic energy preservation methods. In: Proceedings of the ICOSAHOM-2019, July 1–5, Paris, France (2019)
Acknowledgements
Financial support from the NASA TTT/RCA program for the second author is gratefully acknowledged. The authors are grateful to Dr. Alan Wray of NASA Ames Research Center for the numerous invaluable discussions throughout the course of this work.
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Sjögreen, B., Yee, H.C. Construction of Conservative Numerical Fluxes for the Entropy Split Method. Commun. Appl. Math. Comput. 5, 653–678 (2023). https://doi.org/10.1007/s42967-020-00111-4
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DOI: https://doi.org/10.1007/s42967-020-00111-4