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

doi:10.1007/s42967-021-00148-z

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (3): 880-903.doi: 10.1007/s42967-021-00148-z

• ORIGINAL PAPER • 上一篇下一篇

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

Hendrik Ranocha^1,2, Gregor J. Gassner³

1. Applied Mathematics Münster, University of Münster, 48149 Münster, Germany;
2. Computer Electrical and Mathematical Science and Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia;
3. Department of Mathematics and Computer Science, Center for Data and Simulation Science, University of Cologne, Cologne, Germany

收稿日期:2020-09-28 修回日期:2021-04-13 出版日期:2022-09-20 发布日期:2022-07-04
通讯作者: Hendrik Ranocha,E-mail:mail@ranocha.de;Gregor J. Gassner,E-mail:ggassner@math.uni-koeln.de E-mail:mail@ranocha.de
基金资助:
Research reported in this publication was supported by the King Abdullah University of Science and Technology (KAUST). Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure. Gregor Gassner 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.

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

Hendrik Ranocha^1,2, Gregor J. Gassner³

1. Applied Mathematics Münster, University of Münster, 48149 Münster, Germany;
2. Computer Electrical and Mathematical Science and Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia;
3. Department of Mathematics and Computer Science, Center for Data and Simulation Science, University of Cologne, Cologne, Germany

Received:2020-09-28 Revised:2021-04-13 Online:2022-09-20 Published:2022-07-04
Contact: Hendrik Ranocha,E-mail:mail@ranocha.de;Gregor J. Gassner,E-mail:ggassner@math.uni-koeln.de E-mail:mail@ranocha.de
Supported by:
Research reported in this publication was supported by the King Abdullah University of Science and Technology (KAUST). Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure. Gregor Gassner 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.

摘要/Abstract

摘要： Recently, 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).

关键词: Entropy conservation, Kinetic energy preservation, Pressure equilibrium preservation, Compressible Euler equations, Local linear stability, Summation-by-parts

Abstract: Recently, 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).

Key words: Entropy conservation, Kinetic energy preservation, Pressure equilibrium preservation, Compressible Euler equations, Local linear stability, Summation-by-parts

中图分类号:

Hendrik Ranocha, Gregor J. Gassner. Preventing Pressure Oscillations Does Not Fix Local Linear Stability Issues of Entropy-Based Split-Form High-Order Schemes[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 880-903.

参考文献

[1] 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]
[2] 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
[3] 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
[4] 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
[5] 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
[6] Chen, H.: Means generated by an integral. Math. Mag. 78(5), 397–399 (2005). https://doi.org/10.2307/30044201
[7] 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
[8] 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
[9] 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
[10] 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
[11] 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
[12] 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
[13] 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]
[14] 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
[15] 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
[16] 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
[17] 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
[18] 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
[19] 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
[20] Hunter, J.D.: Matplotlib: a 2D graphics environment. Comput. Sci. Eng. 9(3), 90–95 (2007). https://doi.org/10.1109/MCSE.2007.55
[21] 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
[22] 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
[23] 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)
[24] 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
[25] 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)
[26] 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
[27] 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
[28] 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
[29] 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
[30] 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
[31] 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
[32] Ranocha, H.: Comparison of some entropy conservative numerical fluxes for the Euler equations. J. Sci. Comput. arXiv:1701.02264 [math.NA]
[33] Ranocha, H.: Generalised summation-by-parts operators and entropy stability of numerical methods for hyperbolic balance laws. Ph.D. thesis, TU Braunschweig (2018)
[34] 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
[35] Ranocha, H.: On strong stability of explicit Runge-Kutta methods for nonlinear semibounded operators. IMA J. Numer. Anal. (2020). . [math.NA] 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]
[36] 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). . 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
[37] 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]
[38] 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]
[39] Ranocha, H., Öffner, P., Sonar, T.: Summation-by-parts operators for correction procedure via reconstruction. J. Comput. Phys. arXiv:1511.02052 [math.NA]
[40] Revels, J., Lubin, M., Papamarkou, T.: Forward-mode automatic differentiation in Julia (2016). arXiv:1607.07892 [cs.MS]
[41] 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]
[42] 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). . 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
[43] 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]
[44] 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
[45] 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
[46] 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
[47] 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
[48] Strand, B.: Summation by parts for finite difference approximations for

\begin{document}$ d/dx $\end{document}

. J. Comput. Phys. 110(1), 47–67 (1994). https://doi.org/10.1006/jcph.1994.1005
[49] 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
[50] 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
[51] 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
[52] 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
[53] 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

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

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

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	Xiaozhou Li. How to Design a Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 759-782.
[2]	Poorvi Shukla, J. J. W. van der Vegt. A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 904-944.
[3]	Mohammed Homod Hashim, Akil J. Harfash. Finite Element Analysis of Attraction-Repulsion Chemotaxis System. Part I: Space Convergence[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 1011-1056.
[4]	Mohammed Homod Hashim, Akil J. Harfash. Finite Element Analysis of Attraction-Repulsion Chemotaxis System. Part II: Time Convergence, Error Analysis and Numerical Results[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 1057-1104.
[5]	Jiawei Sun, Shusen Xie, Yulong Xing. Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 381-416.
[6]	Rami Masri, Charles Puelz, Beatrice Riviere. A Discontinuous Galerkin Method for Blood Flow and Solute Transport in One-Dimensional Vessel Networks[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 500-529.
[7]	Xue Hong, Yinhua Xia. Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Methods for KdV Type Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 530-562.
[8]	Erik Burman, Omar Duran, Alexandre Ern. Hybrid High-Order Methods for the Acoustic Wave Equation in the Time Domain[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 597-633.
[9]	Hongjuan Zhang, Boying Wu, Xiong Meng. A Local Discontinuous Galerkin Method with Generalized Alternating Fluxes for 2D Nonlinear Schrödinger Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 84-107.
[10]	Qi Tao, Yan Xu, Xiaozhou Li. Negative Norm Estimates for Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Nonlinear Hyperbolic Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 250-270.
[11]	Haijin Wang, Qiang Zhang. The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Difusion Problems[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 271-292.
[12]	Yuan Xu, Qiang Zhang. Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 319-352.
[13]	Jie Du, Eric Chung, Yang Yang. Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 353-379.
[14]	Junhong Tian, Hengfei Ding. A Note on Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Difusion Equation[J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 571-584.
[15]	Zheng Sun, Chi-Wang Shu. Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity[J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 671-700.