Preventing Pressure Oscillations Does Not Fix Local Linear Stability Issues of Entropy-Based Split-Form High-Order Schemes

Communications on Applied Mathematics and Computation - Tập 4 Số 3 - Trang 880-903 - 2022
Hendrik Ranocha1, Gregor J. Gassner2
1Applied Mathematics Münster, University of Münster, 48149 , Münster, Germany
2Department of Mathematics and Computer Science, Center for Data and Simulation Science, University of Cologne, Cologne, Germany

Tóm tắt

AbstractRecently, it was discovered that the entropy-conserving/dissipative high-order split-form discontinuous Galerkin discretizations have robustness issues when trying to solve the simple density wave propagation example for the compressible Euler equations. The issue is related to missing local linear stability, i.e., the stability of the discretization towards perturbations added to a stable base flow. This is strongly related to an anti-diffusion mechanism, that is inherent in entropy-conserving two-point fluxes, which are a key ingredient for the high-order discontinuous Galerkin extension. In this paper, we investigate if pressure equilibrium preservation is a remedy to these recently found local linear stability issues of entropy-conservative/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations. Pressure equilibrium preservation describes the property of a discretization to keep pressure and velocity constant for pure density wave propagation. We present the full theoretical derivation, analysis, and show corresponding numerical results to underline our findings. In addition, we characterize numerical fluxes for the Euler equations that are entropy-conservative, kinetic-energy-preserving, pressure-equilibrium-preserving, and have a density flux that does not depend on the pressure. The source code to reproduce all numerical experiments presented in this article is available online (https://doi.org/10.5281/zenodo.4054366).

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Tài liệu tham khảo

Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59(1), 65–98 (2017). arXiv:1411.1607 [cs.MS]

Carpenter, M.H., Fisher, T.C., Nielsen, E.J., Frankel, S.H.: Entropy stable spectral collocation schemes for the Navier-Stokes equations: discontinuous interfaces. SIAM J. Sci. Comput. 36(5), B835–B867 (2014). https://doi.org/10.1137/130932193

Carpenter, M.H., Parsani, M., Fisher, T.C., Nielsen, E.J.: Towards an entropy stable spectral element framework for computational fluid dynamics. In: 54th AIAA Aerospace Sciences Meeting. American Institute of Aeronautics and Astronautics (2016). https://doi.org/10.2514/6.2016-1058

Chan, J.: On discretely entropy conservative and entropy stable discontinuous Galerkin methods. J. Comput. Phys. 362, 346–374 (2018). https://doi.org/10.1016/j.jcp.2018.02.033

Chan, J., Fernández, D.C.D.R., Carpenter, M.H.: Efficient entropy stable Gauss collocation methods. SIAM J. Sci. Comput. 41(5), A2938–A2966 (2019). https://doi.org/10.1137/18M1209234

Chen, H.: Means generated by an integral. Math. Mag. 78(5), 397–399 (2005). https://doi.org/10.2307/30044201

Chen, T., Shu, C.-W.: Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345, 427–461 (2017). https://doi.org/10.1016/j.jcp.2017.05.025

Derigs, D., Winters, A.R., Gassner, G.J., Walch, S.: A novel averaging technique for discrete entropy-stable dissipation operators for ideal MHD. J. Comput. Phys. 330, 624–632 (2017). https://doi.org/10.1016/j.jcp.2016.10.055

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). https://doi.org/10.1016/j.compfluid.2014.02.016

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). https://doi.org/10.1016/j.jcp.2013.06.014

Flad, D., Gassner, G.: On the use of kinetic energy preserving DG-schemes for large eddy simulation. J. Comput. Phys. 350, 782–795 (2017). https://doi.org/10.1016/j.jcp.2017.09.004

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). https://doi.org/10.1137/120890144

Gassner, G.J., Svärd, M., Hindenlang, F.J.: Stability issues of entropy-stable and/or split-form high-order schemes (2020). arXiv:2007.09026 [math.NA]

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). https://doi.org/10.1016/j.jcp.2016.09.013

Harten, A.: On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49(1), 151–164 (1983). https://doi.org/10.1016/0021-9991(83)90118-3

Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983). https://doi.org/10.1137/1025002

Hicken, J.E.: Entropy-stable, high-order summation-by-parts discretizations without interface penalties. J. Sci. Comput. 82(2), 50 (2020). https://doi.org/10.1007/s10915-020-01154-8

Hicken, J.E., Fernández, D.C.D.R., Zingg, D.W.: Multidimensional summation-by-parts operators: general theory and application to simplex elements. SIAM J. Sci. Comput. 38(4), A1935–A1958 (2016). https://doi.org/10.1137/15M1038360

Hughes, T.J.R., Franca, L.P., Mallet, M.: A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. Comput. Methods Appl. Mech. Eng. 54(2), 223–234 (1986). https://doi.org/10.1016/0045-7825(86)90127-1

Hunter, J.D.: Matplotlib: a 2D graphics environment. Comput. Sci. Eng. 9(3), 90–95 (2007). https://doi.org/10.1109/MCSE.2007.55

Ismail, F., Roe, P.L.: Affordable, entropy-consistent Euler flux functions II: entropy production at shocks. J. Comput. Phys. 228(15), 5410–5436 (2009). https://doi.org/10.1016/j.jcp.2009.04.021

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). https://doi.org/10.1007/s10915-007-9172-6

Kennedy, C.A., Carpenter, M.H.: Fourth order 2N-storage Runge-Kutta schemes. Technical Memorandum NASA-TM-109112, NASA, NASA Langley Research Center, Hampton (1994)

Klose, B.F., Jacobs, G.B., Kopriva, D.A.: Assessing standard and kinetic energy conserving volume fluxes in discontinuous Galerkin formulations for marginally resolved Navier-Stokes flows. Comput. Fluids (2020). https://doi.org/10.1016/j.compfluid.2020.104557

Kreiss, H.O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 195–212. Academic Press, New York (1974)

Kuya, Y., Totani, K., Kawai, S.: Kinetic energy and entropy preserving schemes for compressible flows by split convective forms. J. Comput. Phys. 375, 823–853 (2018). https://doi.org/10.1016/j.jcp.2018.08.058

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). https://doi.org/10.1137/S003614290240069X

Nordström, J., Björck, M.: Finite volume approximations and strict stability for hyperbolic problems. Appl. Numer. Math. 38(3), 237–255 (2001). https://doi.org/10.1016/S0168-9274(01)00027-7

Nordström, J., Forsberg, K., Adamsson, C., Eliasson, P.: Finite volume methods, unstructured meshes and strict stability for hyperbolic problems. Appl. Numer. Math. 45(4), 453–473 (2003). https://doi.org/10.1016/S0168-9274(02)00239-8

Parsani, M., Carpenter, M.H., Nielsen, E.J.: Entropy stable discontinuous interfaces coupling for the three-dimensional compressible Navier-Stokes equations. J. Comput. Phys. 290, 132–138 (2015). https://doi.org/10.1016/j.jcp.2015.02.042

Parsani, M., Carpenter, M.H., Nielsen, E.J.: Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations. J. Comput. Phys. 292, 88–113 (2015). https://doi.org/10.1016/j.jcp.2015.03.026

Ranocha, H.: Comparison of some entropy conservative numerical fluxes for the Euler equations. J. Sci. Comput. arXiv:1701.02264 [math.NA]

Ranocha, H.: Generalised summation-by-parts operators and entropy stability of numerical methods for hyperbolic balance laws. Ph.D. thesis, TU Braunschweig (2018)

Ranocha, H.: Entropy conserving and kinetic energy preserving numerical methods for the Euler equations using summation-by-parts operators. In: Sherwin S.J., Moxey, D., Peiró, J., Vincent, P.E., Schwab, C. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018, Lecture Notes in Computational Science and Engineering, vol. 134, pp. 525–535. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-39647-3_42

Ranocha, H.: On strong stability of explicit Runge-Kutta methods for nonlinear semibounded operators. IMA J. Numer. Anal. (2020). https://doi.org/10.1093/imanum/drz070. arXiv:1811.11601 [math.NA]

Ranocha, H., Gassner, G.J.: Reproducibility: preventing pressure oscillations does not fix local linear stability issues of entropy-based split-form high-order schemes (2020). https://github.com/trixi-framework/paper-EC-KEP-PEP. https://doi.org/10.5281/zenodo.4054366

Ranocha, H., Ketcheson, D.I.: Energy stability of explicit Runge-Kutta methods for nonautonomous or nonlinear problems. SIAM J. Numer. Anal. arXiv:1909.13215 [math.NA]

Ranocha, H., Mitsotakis, D., Ketcheson, D.I.: A broad class of conservative numerical methods for dispersive wave equations. Commun. Comput. Phys. arXiv:2006.14802 [math.NA]

Ranocha, H., Öffner, P., Sonar, T.: Summation-by-parts operators for correction procedure via reconstruction. J. Comput. Phys. arXiv:1511.02052 [math.NA]

Revels, J., Lubin, M., Papamarkou, T.: Forward-mode automatic differentiation in Julia (2016). arXiv:1607.07892 [cs.MS]

Rojas, D., Boukharfane, R., Dalcin, L., Fernández, D.C.D.R., Ranocha, H., Keyes, D.E., Parsani, M.: On the robustness and performance of entropy stable discontinuous collocation methods. J. Comput. Phys. arXiv:1911.10966 [math.NA]

Schlottke-Lakemper, M., Gassner, G.J., Ranocha, H., Winters, A.R.: Trixi.jl: a tree-based numerical simulation framework for hyperbolic PDEs written in Julia (2020). https://github.com/trixi-framework/Trixi.jl. https://doi.org/10.5281/zenodo.3996439

Schlottke-Lakemper, M., Winters, A.R., Ranocha, H., Gassner, G.J.: A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics (2020). arXiv:2008.10593 [math.NA]

Shima, N., Kuya, Y., Tamaki, Y., Kawai, S.: Preventing spurious pressure oscillations in split convective form discretization for compressible flows. J. Comput. Phys. (2020). https://doi.org/10.1016/j.jcp.2020.110060

Sjögreen, B., Yee, H.: High order entropy conservative central schemes for wide ranges of compressible gas dynamics and MHD flows. J. Comput. Phys. 364, 153–185 (2018). https://doi.org/10.1016/j.jcp.2018.02.003

Sjögreen, B., Yee, H.C.: On skew-symmetric splitting and entropy conservation schemes for the Euler equations. In: Kreiss, G., Lötstedt, P., Målqvist, A., Neytcheva, M. (eds) Numerical Mathematics and Advanced Applications 2009: Proceedings of ENUMATH 2009, the 8th European Conference on Numerical Mathematics and Advanced Applications, Uppsala, July 2009, pp. 817–827. Springer, Berlin (2010). https://doi.org/10.1007/978-3-642-11795-4_88

Sjögreen, B., Yee, H.C., Kotov, D.: Skew-symmetric splitting and stability of high order central schemes. J. Phys. Conf. Ser. 837, 012019 (2017). https://doi.org/10.1088/1742-6596/837/1/012019

Strand, B.: Summation by parts for finite difference approximations for $$d/dx$$. J. Comput. Phys. 110(1), 47–67 (1994). https://doi.org/10.1006/jcph.1994.1005

Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014). https://doi.org/10.1016/j.jcp.2014.02.031

Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49(179), 91–103 (1987). https://doi.org/10.1090/S0025-5718-1987-0890255-3

Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003). https://doi.org/10.1017/S0962492902000156

Vasil, G., Brown, B., Burns, K., Lecoanet, D., McCourt, M., Oishi, J., O’Leary, R., Quataert, E., Stone, J.: A validated non-linear Kelvin-Helmholtz benchmark for numerical hydrodynamics. Mon. Not. R. Astron. Soc. 455(4), 4274–4288 (2016). https://doi.org/10.1093/mnras/stv2564

Winters, A.R., Moura, R.C., Mengaldo, G., Gassner, G.J., Walch, S., Peiro, J., Sherwin, S.J.: A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations. J. Comput. Phys. 372, 1–21 (2018). https://doi.org/10.1016/j.jcp.2018.06.016