Towards a fully well-balanced and entropy-stable scheme for the Euler equations with gravity: preserving isentropic steady solutions

Berthon, Christophe; Michel-Dansac, Victor; Thomann, Andrea

doi:10.1090/mcom/4088

Towards a fully well-balanced and entropy-stable scheme for the Euler equations with gravity: preserving isentropic steady solutions
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by Christophe Berthon, Victor Michel-Dansac and Andrea Thomann;

Math. Comp.

DOI: https://doi.org/10.1090/mcom/4088

Published electronically: April 17, 2025

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Abstract:

The present work concerns the derivation of a numerical scheme to approximate weak solutions of the Euler equations with a gravitational source term. The designed scheme is proved to be fully well-balanced since it is able to exactly preserve all moving equilibrium solutions, as well as the corresponding steady solutions at rest obtained when the velocity vanishes. Moreover, the proposed scheme is entropy-stable since it satisfies all fully discrete entropy inequalities. In addition, in order to satisfy the required admissibility of the approximate solutions, the positivity of both approximate density and pressure is established. Several numerical experiments attest the relevance of the developed numerical method. An extension to two-dimensional problems is given, applying the one-dimensional framework direction by direction on Cartesian grids.

References

Emmanuel Audusse, François Bouchut, Marie-Odile Bristeau, Rupert Klein, and Benoît Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput. 25 (2004), no. 6, 2050–2065. MR 2086830, DOI 10.1137/S1064827503431090
Alfredo Bermudez and Ma. Elena Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids 23 (1994), no. 8, 1049–1071. MR 1314237, DOI 10.1016/0045-7930(94)90004-3
Christophe Berthon, Numerical approximations of the 10-moment Gaussian closure, Math. Comp. 75 (2006), no. 256, 1809–1831. MR 2240636, DOI 10.1090/S0025-5718-06-01860-6
Christophe Berthon, Solène Bulteau, Françoise Foucher, Meissa M’Baye, and Victor Michel-Dansac, A very easy high-order well-balanced reconstruction for hyperbolic systems with source terms, SIAM J. Sci. Comput. 44 (2022), no. 4, A2506–A2535. MR 4469516, DOI 10.1137/21M1429230
Christophe Berthon and Christophe Chalons, A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations, Math. Comp. 85 (2016), no. 299, 1281–1307. MR 3454365, DOI 10.1090/mcom3045
Christophe Berthon and Françoise Foucher, Efficient well-balanced hydrostatic upwind schemes for shallow-water equations, J. Comput. Phys. 231 (2012), no. 15, 4993–5015. MR 2929930, DOI 10.1016/j.jcp.2012.02.031
Christophe Berthon, Meissa M’baye, Minh Hoang Le, and Diaraf Seck, A well-defined moving steady states capturing Godunov-type scheme for shallow-water model, Int. J. Finite Vol. 15 (2020), 33. MR 4250070
Christophe Berthon and Victor Michel-Dansac, A fully well-balanced hydrodynamic reconstruction, J. Numer. Math. 32 (2024), no. 3, 275–299. MR 4796048, DOI 10.1515/jnma-2023-0065
François Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004. MR 2128209, DOI 10.1007/b93802
François Bouchut and Tomás Morales de Luna, A subsonic-well-balanced reconstruction scheme for shallow water flows, SIAM J. Numer. Anal. 48 (2010), no. 5, 1733–1758. MR 2733096, DOI 10.1137/090758416
Jolene Britton and Yulong Xing, High order still-water and moving-water equilibria preserving discontinuous Galerkin methods for the Ripa model, J. Sci. Comput. 82 (2020), no. 2, Paper No. 30, 37. MR 4055242, DOI 10.1007/s10915-020-01134-y
Patricia Cargo and Alain Yves LeRoux, Un schéma équilibre adapté au modèle d’atmosphère avec termes de gravité, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 1, 73–76 (French, with English and French summaries). MR 1260539
Manuel Castro, José M. Gallardo, Juan A. López-García, and Carlos Parés, Well-balanced high order extensions of Godunov’s method for semilinear balance laws, SIAM J. Numer. Anal. 46 (2008), no. 2, 1012–1039. MR 2383221, DOI 10.1137/060674879
Manuel J. Castro, Alberto Pardo Milanés, and Carlos Parés, Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique, Math. Models Methods Appl. Sci. 17 (2007), no. 12, 2055–2113. MR 2371563, DOI 10.1142/S021820250700256X
Christophe Chalons and Jean-François Coulombel, Relaxation approximation of the Euler equations, J. Math. Anal. Appl. 348 (2008), no. 2, 872–893. MR 2446042, DOI 10.1016/j.jmaa.2008.07.034
Praveen Chandrashekar and Christian Klingenberg, A second order well-balanced finite volume scheme for Euler equations with gravity, SIAM J. Sci. Comput. 37 (2015), no. 3, B382–B402. MR 3348134, DOI 10.1137/140984373
Guoxian Chen and Sebastian Noelle, A new hydrostatic reconstruction scheme based on subcell reconstructions, SIAM J. Numer. Anal. 55 (2017), no. 2, 758–784. MR 3631390, DOI 10.1137/15M1053074
Yuanzhen Cheng, Alina Chertock, Michael Herty, Alexander Kurganov, and Tong Wu, A new approach for designing moving-water equilibria preserving schemes for the shallow water equations, J. Sci. Comput. 80 (2019), no. 1, 538–554. MR 3954453, DOI 10.1007/s10915-019-00947-w
Frédéric Coquel and Benoît Perthame, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics, SIAM J. Numer. Anal. 35 (1998), no. 6, 2223–2249. MR 1655844, DOI 10.1137/S0036142997318528
Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 4th ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2016. MR 3468916, DOI 10.1007/978-3-662-49451-6
Vivien Desveaux, Markus Zenk, Christophe Berthon, and Christian Klingenberg, A well-balanced scheme to capture non-explicit steady states in the Euler equations with gravity, Internat. J. Numer. Methods Fluids 81 (2016), no. 2, 104–127. MR 3491831, DOI 10.1002/fld.4177
Steven Diot, Stéphane Clain, and Raphaël Loubère, Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials, Comput. & Fluids 64 (2012), 43–63. MR 2982757, DOI 10.1016/j.compfluid.2012.05.004
Eduard Feireisl, Mária Lukáčová-Medvid’ová, Hana Mizerová, and Bangwei She, Numerical analysis of compressible fluid flows, MS&A. Modeling, Simulation and Applications, vol. 20, Springer, Cham, [2021] ©2021. MR 4390192, DOI 10.1007/978-3-030-73788-7
Emmanuel Franck and Laura S. Mendoza, Finite volume scheme with local high order discretization of the hydrostatic equilibrium for the Euler equations with external forces, J. Sci. Comput. 69 (2016), no. 1, 314–354. MR 3542802, DOI 10.1007/s10915-016-0199-4
Emmanuel Franck, Victor Michel-Dansac, and Laurent Navoret, Approximately well-balanced discontinuous Galerkin methods using bases enriched with physics-informed neural networks, J. Comput. Phys. 512 (2024), Paper No. 113144, 32. MR 4752817, DOI 10.1016/j.jcp.2024.113144
F. G. Fuchs, A. D. McMurry, S. Mishra, N. H. Risebro, and K. Waagan, High order well-balanced finite volume schemes for simulating wave propagation in stratified magnetic atmospheres, J. Comput. Phys. 229 (2010), no. 11, 4033–4058. MR 2609765, DOI 10.1016/j.jcp.2010.01.038
José M. Gallardo, Carlos Parés, and Manuel Castro, On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, J. Comput. Phys. 227 (2007), no. 1, 574–601. MR 2361537, DOI 10.1016/j.jcp.2007.08.007
Beatrice Ghitti, Christophe Berthon, Minh Hoang Le, and Eleuterio F. Toro, A fully well-balanced scheme for the 1D blood flow equations with friction source term, J. Comput. Phys. 421 (2020), 109750, 33. MR 4135562, DOI 10.1016/j.jcp.2020.109750
Edwige Godlewski and Pierre-Arnaud Raviart, Hyperbolic systems of conservation laws, Mathématiques & Applications (Paris) [Mathematics and Applications], vol. 3/4, Ellipses, Paris, 1991. MR 1304494
I. Gómez-Bueno, M. J. Castro Díaz, C. Parés, and G. Russo, Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws, Mathematics 9 (2021), no. 15, 1799.
L. Gosse, A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms, Comput. Math. Appl. 39 (2000), no. 9-10, 135–159. MR 1753567, DOI 10.1016/S0898-1221(00)00093-6
Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001), no. 1, 89–112. MR 1854647, DOI 10.1137/S003614450036757X
J. M. Greenberg and A. Y. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal. 33 (1996), no. 1, 1–16. MR 1377240, DOI 10.1137/0733001
L. Grosheintz-Laval and R. Käppeli, High-order well-balanced finite volume schemes for the Euler equations with gravitation, J. Comput. Phys. 378 (2019), 324–343. MR 3881590, DOI 10.1016/j.jcp.2018.11.018
Amiram Harten, Peter D. Lax, and Bram van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983), no. 1, 35–61. MR 693713, DOI 10.1137/1025002
Shi Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms, M2AN Math. Model. Numer. Anal. 35 (2001), no. 4, 631–645. MR 1862872, DOI 10.1051/m2an:2001130
Shi Jin and C. David Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys. 126 (1996), no. 2, 449–467. MR 1404381, DOI 10.1006/jcph.1996.0149
R. Käppeli and S. Mishra, Well-balanced schemes for the Euler equations with gravitation, J. Comput. Phys. 259 (2014), 199–219. MR 3148566, DOI 10.1016/j.jcp.2013.11.028
C. Klingenberg, G. Puppo, and M. Semplice, Arbitrary order finite volume well-balanced schemes for the Euler equations with gravity, SIAM J. Sci. Comput. 41 (2019), no. 2, A695–A721. MR 3922235, DOI 10.1137/18M1196704
Jun Luo, Kun Xu, and Na Liu, A well-balanced symplecticity-preserving gas-kinetic scheme for hydrodynamic equations under gravitational field, SIAM J. Sci. Comput. 33 (2011), no. 5, 2356–2381. MR 2837536, DOI 10.1137/100803699
Victor Michel-Dansac, Christophe Berthon, Stéphane Clain, and Françoise Foucher, A well-balanced scheme for the shallow-water equations with topography, Comput. Math. Appl. 72 (2016), no. 3, 568–593. MR 3521058, DOI 10.1016/j.camwa.2016.05.015
Victor Michel-Dansac, Christophe Berthon, Stéphane Clain, and Françoise Foucher, A well-balanced scheme for the shallow-water equations with topography or Manning friction, J. Comput. Phys. 335 (2017), 115–154. MR 3612493, DOI 10.1016/j.jcp.2017.01.009
V. Michel-Dansac, C. Berthon, S. Clain, and F. Foucher, A two-dimensional high-order well-balanced scheme for the shallow water equations with topography and Manning friction, Comput. & Fluids 230 (2021), 105152.
Carlos Parés and Manuel Castro, On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems, M2AN Math. Model. Numer. Anal. 38 (2004), no. 5, 821–852. MR 2104431, DOI 10.1051/m2an:2004041
Benoit Perthame and Chi-Wang Shu, On positivity preserving finite volume schemes for Euler equations, Numer. Math. 73 (1996), no. 1, 119–130. MR 1379283, DOI 10.1007/s002110050187
B. Schmidtmann, B. Seibold, and M. Torrilhon, Relations between WENO3 and third-order limiting in finite volume methods, J. Sci. Comput. 68 (2016), no. 2, 624–652. MR 3519195, DOI 10.1007/s10915-015-0151-z
Eitan Tadmor, A minimum entropy principle in the gas dynamics equations, Appl. Numer. Math. 2 (1986), no. 3-5, 211–219. MR 863987, DOI 10.1016/0168-9274(86)90029-2
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
Andrea Thomann, Gabriella Puppo, and Christian Klingenberg, An all speed second order well-balanced IMEX relaxation scheme for the Euler equations with gravity, J. Comput. Phys. 420 (2020), 109723, 25. MR 4128629, DOI 10.1016/j.jcp.2020.109723
Andrea Thomann, Markus Zenk, and Christian Klingenberg, A second-order positivity-preserving well-balanced finite volume scheme for Euler equations with gravity for arbitrary hydrostatic equilibria, Internat. J. Numer. Methods Fluids 89 (2019), no. 11, 465–482. MR 3934857, DOI 10.1002/fld.4703
Eleuterio F. Toro, Riemann solvers and numerical methods for fluid dynamics, 3rd ed., Springer-Verlag, Berlin, 2009. A practical introduction. MR 2731357, DOI 10.1007/b79761
E. F. Toro, M. Spruce, and W. Speares, Restoration of the contact surface in the HLL-Riemann solver, Shock Waves 4 (1994), no. 1, 25–34.
B. van Leer, Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov’s method, J. Comput. Phys. 32 (1979), 101–136.
J. Vides, B. Braconnier, E. Audit, C. Berthon, and B. Nkonga, A Godunov-type solver for the numerical approximation of gravitational flows, Commun. Comput. Phys. 15 (2014), no. 1, 46–75. MR 3094198, DOI 10.4208/cicp.060712.210313a
Yulong Xing, Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium. part A, J. Comput. Phys. 257 (2014), no. part A, 536–553. MR 3129548, DOI 10.1016/j.jcp.2013.10.010
Yulong Xing and Chi-Wang Shu, High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields, J. Sci. Comput. 54 (2013), no. 2-3, 645–662. MR 3011375, DOI 10.1007/s10915-012-9585-8
Kun Xu, A well-balanced gas-kinetic scheme for the shallow-water equations with source terms, J. Comput. Phys. 178 (2002), no. 2, 533–562. MR 1884320, DOI 10.1006/jcph.2002.7040
Kun Xu, Jun Luo, and Songze Chen, A well-balanced kinetic scheme for gas dynamic equations under gravitational field, Adv. Appl. Math. Mech. 2 (2010), no. 2, 200–210. MR 2606447, DOI 10.4208/aamm.09-m0964

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Towards a fully well-balanced and entropy-stable scheme for the Euler equations with gravity: preserving isentropic steady solutions
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Abstract: