Strictly convex entropy and entropy stable schemes for reactive Euler equations
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Abstract:
This paper presents entropy analysis and entropy stable (ES) finite difference schemes for the reactive Euler equations with chemical reactions. For such equations we point out that the thermodynamic entropy is no longer strictly convex. To address this issue, we propose a strictly convex entropy function by adding an extra term to the thermodynamic entropy. Thanks to the strict convexity of the proposed entropy, the Roe-type dissipation operator in terms of the entropy variables can be formulated. Furthermore, we construct two sets of second-order entropy preserving (EP) numerical fluxes for the reactive Euler equations. Based on the EP fluxes and the Roe-type dissipation operators, high-order EP/ES fluxes are derived. Numerical experiments validate the designed accuracy and good performance of our schemes for smooth and discontinuous initial value problems. The entropy decrease of ES schemes is verified as well.References
- Uri M. Ascher, Steven J. Ruuth, and Raymond J. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math. 25 (1997), no. 2-3, 151–167. Special issue on time integration (Amsterdam, 1996). MR 1485812, DOI 10.1016/S0168-9274(97)00056-1
- Timothy J. Barth, Numerical methods for gasdynamic systems on unstructured meshes, An introduction to recent developments in theory and numerics for conservation laws (Freiburg/Littenweiler, 1997) Lect. Notes Comput. Sci. Eng., vol. 5, Springer, Berlin, 1999, pp. 195–285. MR 1731618, DOI 10.1007/978-3-642-58535-7_{5}
- Biswarup Biswas and Ritesh Kumar Dubey, Low dissipative entropy stable schemes using third order WENO and TVD reconstructions, Adv. Comput. Math. 44 (2018), no. 4, 1153–1181. MR 3842951, DOI 10.1007/s10444-017-9576-2
- Anne Bourlioux, Andrew J. Majda, and Victor Roytburd, Theoretical and numerical structure for unstable one-dimensional detonations, SIAM J. Appl. Math. 51 (1991), no. 2, 303–343. MR 1095022, DOI 10.1137/0151016
- Praveen Chandrashekar, Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations, Commun. Comput. Phys. 14 (2013), no. 5, 1252–1286. MR 3079107, DOI 10.4208/cicp.170712.010313a
- Praveen Chandrashekar and Christian Klingenberg, Entropy stable finite volume scheme for ideal compressible MHD on 2-D Cartesian meshes, SIAM J. Numer. Anal. 54 (2016), no. 2, 1313–1340. MR 3490501, DOI 10.1137/15M1013626
- Gui-Qiang Chen and David H. Wagner, Global entropy solutions to exothermically reacting, compressible Euler equations, J. Differential Equations 191 (2003), no. 2, 277–322. MR 1978380, DOI 10.1016/S0022-0396(03)00027-5
- Tianheng Chen and Chi-Wang Shu, Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws, J. Comput. Phys. 345 (2017), 427–461. MR 3667622, DOI 10.1016/j.jcp.2017.05.025
- J. F. Clarke, S. Karni, J. J. Quirk, P. L. Roe, L. G. Simmonds, and E. F. Toro, Numerical computation of two-dimensional unsteady detonation waves in high energy solids, J. Comput. Phys. 106 (1993), no. 2, 215–233.
- Phillip Colella, Andrew Majda, and Victor Roytburd, Theoretical and numerical structure for reacting shock waves, SIAM J. Sci. Statist. Comput. 7 (1986), no. 4, 1059–1080. MR 857783, DOI 10.1137/0907073
- Junming Duan and Huazhong Tang, High-order accurate entropy stable finite difference schemes for one- and two-dimensional special relativistic hydrodynamics, Adv. Appl. Math. Mech. 12 (2020), no. 1, 1–29. MR 4048875, DOI 10.4208/aamm.oa-2019-0124
- W. Fickett, Detonations in miniature, University of California Press, Berkeley, 1985.
- W. Fickett and W. W. Wood, Flow calculations for pulsating one dimensional detonations, Phys. Fluids 9 (1966), no. 5, 903–916.
- Travis C. Fisher and Mark H. Carpenter, High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains, J. Comput. Phys. 252 (2013), 518–557. MR 3101520, DOI 10.1016/j.jcp.2013.06.014
- Ulrik S. Fjordholm, Siddhartha Mishra, and Eitan Tadmor, Energy preserving and energy stable schemes for the shallow water equations, Foundations of computational mathematics, Hong Kong 2008, London Math. Soc. Lecture Note Ser., vol. 363, Cambridge Univ. Press, Cambridge, 2009, pp. 93–139. MR 2562498
- Ulrik S. Fjordholm, Siddhartha Mishra, and Eitan Tadmor, Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws, SIAM J. Numer. Anal. 50 (2012), no. 2, 544–573. MR 2914275, DOI 10.1137/110836961
- Ulrik S. Fjordholm, Siddhartha Mishra, and Eitan Tadmor, ENO reconstruction and ENO interpolation are stable, Found. Comput. Math. 13 (2013), no. 2, 139–159. MR 3032678, DOI 10.1007/s10208-012-9117-9
- Lucas Friedrich, Gero Schnücke, Andrew R. Winters, David C. Del Rey Fernández, Gregor J. Gassner, and Mark H. Carpenter, Entropy stable space-time discontinuous Galerkin schemes with summation-by-parts property for hyperbolic conservation laws, J. Sci. Comput. 80 (2019), no. 1, 175–222. MR 3954440, DOI 10.1007/s10915-019-00933-2
- Gregor J. Gassner, A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods, SIAM J. Sci. Comput. 35 (2013), no. 3, A1233–A1253. MR 3048217, DOI 10.1137/120890144
- Guanghui Hu, A numerical study of 2D detonation waves with adaptive finite volume methods on unstructured grids, J. Comput. Phys. 331 (2017), 297–311. MR 3588693, DOI 10.1016/j.jcp.2016.11.041
- Juntao Huang, Weifeng Zhao, and Chi-Wang Shu, A third-order unconditionally positivity-preserving scheme for production-destruction equations with applications to non-equilibrium flows, J. Sci. Comput. 79 (2019), no. 2, 1015–1056. MR 3969000, DOI 10.1007/s10915-018-0881-9
- Farzad Ismail and Philip L. Roe, Affordable, entropy-consistent Euler flux functions. II. Entropy production at shocks, J. Comput. Phys. 228 (2009), no. 15, 5410–5436. MR 2541460, DOI 10.1016/j.jcp.2009.04.021
- Antony Jameson, 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 (2008), no. 2, 188–208. MR 2373037, DOI 10.1007/s10915-007-9172-6
- Doyle D. Knight, Elements of numerical methods for compressible flows, Cambridge Aerospace Series, vol. 19, Cambridge University Press, Cambridge, 2006. MR 2267852, DOI 10.1017/CBO9780511617447
- P. G. Lefloch, J. M. Mercier, and C. Rohde, Fully discrete, entropy conservative schemes of arbitrary order, SIAM J. Numer. Anal. 40 (2002), no. 5, 1968–1992. MR 1950629, DOI 10.1137/S003614290240069X
- Stanley Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal. 21 (1984), no. 2, 217–235. MR 736327, DOI 10.1137/0721016
- Stanley Osher and Eitan Tadmor, On the convergence of difference approximations to scalar conservation laws, Math. Comp. 50 (1988), no. 181, 19–51. MR 917817, DOI 10.1090/S0025-5718-1988-0917817-X
- P. L. Roe, Affordable, entropy consistent flux functions, Eleventh International Conference on Hyperbolic Problems: Theory, Numerics and Applications, 2006.
- Gary A. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys. 27 (1978), no. 1, 1–31. MR 495002, DOI 10.1016/0021-9991(78)90023-2
- Pramod K. Subbareddy and Graham V. Candler, A fully discrete, kinetic energy consistent finite-volume scheme for compressible flows, J. Comput. Phys. 228 (2009), no. 5, 1347–1364. MR 2494220, DOI 10.1016/j.jcp.2008.10.026
- Eitan Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws. I, Math. Comp. 49 (1987), no. 179, 91–103. MR 890255, DOI 10.1090/S0025-5718-1987-0890255-3
- Eitan Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta Numer. 12 (2003), 451–512. MR 2249160, DOI 10.1017/S0962492902000156
- Cheng Wang, Xiangxiong Zhang, Chi-Wang Shu, and Jianguo Ning, Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations, J. Comput. Phys. 231 (2012), no. 2, 653–665. MR 2872096, DOI 10.1016/j.jcp.2011.10.002
- Andrew R. Winters and Gregor J. Gassner, Affordable, entropy conserving and entropy stable flux functions for the ideal MHD equations, J. Comput. Phys. 304 (2016), 72–108. MR 3422404, DOI 10.1016/j.jcp.2015.09.055
- Kailiang Wu and Chi-Wang Shu, Entropy symmetrization and high-order accurate entropy stable numerical schemes for relativistic MHD equations, SIAM J. Sci. Comput. 42 (2020), no. 4, A2230–A2261. MR 4125873, DOI 10.1137/19M1275590
- Wen-An Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal. 172 (2004), no. 2, 247–266. MR 2058165, DOI 10.1007/s00205-003-0304-3
- Xiangxiong Zhang and Chi-Wang Shu, Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms, J. Comput. Phys. 230 (2011), no. 4, 1238–1248. MR 2753359, DOI 10.1016/j.jcp.2010.10.036
- Xiangxiong Zhang and Chi-Wang Shu, Positivity-preserving high order finite difference WENO schemes for compressible Euler equations, J. Comput. Phys. 231 (2012), no. 5, 2245–2258. MR 2876636, DOI 10.1016/j.jcp.2011.11.020
- Weifeng Zhao and Juntao Huang, Boundary treatment of implicit-explicit Runge-Kutta method for hyperbolic systems with source terms, J. Comput. Phys. 423 (2020), 109828, 22. MR 4157647, DOI 10.1016/j.jcp.2020.109828
Additional Information
- Weifeng Zhao
- Affiliation: Department of Applied Mathematics, University of Science and Technology Beijing, Beijing 100083, People’s Republic of China
- ORCID: 0000-0003-4893-1982
- Email: wfzhao@ustb.edu.cn
- Received by editor(s): June 26, 2021
- Published electronically: January 25, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp. 91 (2022), 735-760
- MSC (2020): Primary 65M06; Secondary 76M20
- DOI: https://doi.org/10.1090/mcom/3721
- MathSciNet review: 4379974