×

High-order well-balanced finite volume schemes for the Euler equations with gravitation. (English) Zbl 1416.65266

Summary: A high-order well-balanced scheme for the Euler equations with gravitation is presented. The scheme is able to preserve a spatially high-order accurate discrete representation of isentropic hydrostatic equilibria. It is based on a novel local hydrostatic reconstruction, which, in combination with any standard high-order accurate reconstruction procedure, achieves genuine high-order accuracy for smooth solutions close or away from equilibrium. The resulting scheme is very simple and can be implemented into any existing finite volume code with minimal effort. Moreover, the scheme is not tied to any particular form of the equation of state, which is crucial for example in astrophysical applications. Several numerical experiments were performed with a third-order accurate reconstruction. They demonstrate the robustness and high-order accuracy of the scheme nearby and out of hydrostatic equilibrium.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q31 Euler equations
85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics
76M12 Finite volume methods applied to problems in fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Greenberg, J.; Leroux, A., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33, 1, 1-16 (1996) · Zbl 0876.65064
[2] LeVeque, R. J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, J. Comput. Phys., 146, 1, 346-365 (1998) · Zbl 0931.76059
[3] Audusse, E.; Bouchut, F.; Bristeau, M.-O.; Klein, R.; Perthame, B., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25, 6, 2050-2065 (2004) · Zbl 1133.65308
[4] Noelle, S.; Pankratz, N.; Puppo, G.; Natvig, J. R., Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys., 213, 2, 474-499 (2006) · Zbl 1088.76037
[5] Gosse, L., Computing Qualitatively Correct Approximations of Balance Laws (2013), Springer: Springer Milan · Zbl 1272.65065
[6] LeVeque, R. J.; Bale, D. S., Wave propagation methods for conservation laws with source terms, (Hyperbolic Problems: Theory, Numerics, Applications, Vol. II. Hyperbolic Problems: Theory, Numerics, Applications, Vol. II, Zürich, 1998. Hyperbolic Problems: Theory, Numerics, Applications, Vol. II. Hyperbolic Problems: Theory, Numerics, Applications, Vol. II, Zürich, 1998, Int. Ser. Numer. Math., vol. 130 (1999), Birkhäuser: Birkhäuser Basel), 609-618 · Zbl 0927.35062
[7] Botta, N.; Klein, R.; Langenberg, S.; Lützenkirchen, S., Well balanced finite volume methods for nearly hydrostatic flows, J. Comput. Phys., 196, 2, 539-565 (2004) · Zbl 1109.86304
[8] LeVeque, R. J., A well-balanced path-integral f-wave method for hyperbolic problems with source terms, J. Sci. Comput., 48, 209-226 (2011) · Zbl 1221.65233
[9] Käppeli, R.; Mishra, S., Well-balanced schemes for the Euler equations with gravitation, J. Comput. Phys., 259, 199-219 (2014) · Zbl 1349.76345
[10] Desveaux, V.; Zenk, M.; Berthon, C.; Klingenberg, C., A well-balanced scheme for the Euler equation with a gravitational potential, Springer Proc. Math. Stat., 217-226 (2014) · Zbl 1304.76027
[11] Chandrashekar, P.; Klingenberg, C., A second order well-balanced finite volume scheme for Euler equations with gravity, SIAM J. Sci. Comput., 37, 3, B382-B402 (2015) · Zbl 1320.76078
[12] Käppeli, R.; Mishra, S., A well-balanced finite volume scheme for the Euler equations with gravitation. the exact preservation of hydrostatic equilibrium with arbitrary entropy stratification, Astron. Astrophys., 587, A94 (2016)
[13] Touma, R.; Koley, U.; Klingenberg, C., Well-balanced unstaggered central schemes for the Euler equations with gravitation, SIAM J. Sci. Comput., 38, 5, B773-B807 (2016) · Zbl 1346.76100
[14] Li, G.; Xing, Y., High order finite volume WENO schemes for the Euler equations under gravitational fields, J. Comput. Phys., 316, 145-163 (2016) · Zbl 1349.76356
[15] Käppeli, R., A well-balanced scheme for the Euler equations with gravitation, (Innovative Algorithms and Analysis (2017), Springer International Publishing), 229-241 · Zbl 1371.35206
[16] Chertock, A.; Cui, S.; Kurganov, A.; Özcan, Ş. N.; Tadmor, E., Well-balanced schemes for the Euler equations with gravitation: conservative formulation using global fluxes, J. Comput. Phys., 358, 36-52 (2018) · Zbl 1381.76316
[17] Gaburro, E.; Castro, M. J.; Dumbser, M., Well balanced arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming meshes for the Euler equations of gasdynamics with gravity, Mon. Not. R. Astron. Soc., 477, 2, 2251-2275 (2018)
[18] Xing, Y.; Shu, C.-W., High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields, J. Sci. Comput., 54, 2-3, 645-662 (2013) · Zbl 1260.76022
[19] Li, G.; Xing, Y., Well-balanced finite difference weighted essentially non-oscillatory schemes for the Euler equations with static gravitational fields, Comput. Math. Appl., 75, 6, 2071-2085 (2018) · Zbl 1409.76090
[20] Li, G.; Xing, Y., Well-balanced discontinuous Galerkin methods for the Euler equations under gravitational fields, J. Sci. Comput., 1-21 (2015)
[21] Chandrashekar, P.; Zenk, M., Well-balanced nodal discontinuous Galerkin method for Euler equations with gravity, J. Sci. Comput., 71, 3, 1062-1093 (2017) · Zbl 1462.65180
[22] Li, G.; Xing, Y., Well-balanced discontinuous Galerkin methods with hydrostatic reconstruction for the Euler equations with gravitation, J. Comput. Phys., 352, 445-462 (2018) · Zbl 1375.76089
[23] Fuchs, F.; McMurry, A.; Mishra, S.; Risebro, N.; Waagan, K., High order well-balanced finite volume schemes for simulating wave propagation in stratified magnetic atmospheres, J. Comput. Phys., 229, 11, 4033-4058 (2010) · Zbl 1190.76153
[24] Klingenberg, C.; Puppo, G.; Semplice, M., Arbitrary order finite volume well-balanced schemes for the Euler equations with gravity · Zbl 1412.65125
[25] Vallis, G. K., Atmospheric and Oceanic Fluid Dynamics (2006), Cambridge University Press, Cambridge Books Online
[26] Kippenhahn, R.; Weigert, A.; Weiss, A., Stellar Structure and Evolution, Astronomy and Astrophysics Library · Zbl 1253.85001
[27] Shapiro, S. L.; Teukolsky, S. A.; Holes, Black, White Dwarfs, and Neutron Stars (2007), Wiley-VCH Verlag GmbH
[28] Batten, P.; Clarke, N.; Lambert, C.; Causon, D. M., On the choice of wavespeeds for the hllc Riemann solver, SIAM J. Sci. Comput., 18, 6, 1553-1570 (1997) · Zbl 0992.65088
[29] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction (1997), Springer-Verlag GmbH · Zbl 0888.76001
[30] van Leer, B., Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comput. Phys., 32, 1, 101-136 (1979) · Zbl 1364.65223
[31] Harten, A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49, 3, 357-393 (1983) · Zbl 0565.65050
[32] Colella, P.; Woodward, P. R., The piecewise parabolic method (ppm) for gas-dynamical simulations, J. Comput. Phys., 54, 1, 174-201 (1984) · Zbl 0531.76082
[33] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S. R., Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys., 71, 2, 231-303 (1987) · Zbl 0652.65067
[34] Shu, C.-W., High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev., 51, 1, 82-126 (2009) · Zbl 1160.65330
[35] Cravero, I.; Puppo, G.; Semplice, M.; Visconti CWENO, G., Uniformly accurate reconstructions for balance laws, Math. Comput., 87, 1689-1719 (2017) · Zbl 1412.65102
[36] Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev., 43, 1, 89-112 (2001) · Zbl 0967.65098
[37] Courant, R.; Friedrichs, K.; Lewy, H., Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100, 1, 32-74 (1928) · JFM 54.0486.01
[38] Godlewski, E.; Raviart, P.-A., Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences · Zbl 1063.65080
[39] Hirsch, C., Numerical Computation of Internal and External FlowsThe Fundamentals of Computational Fluid Dynamics, vol. 1 (2007), Butterworth-Heinemann
[40] Laney, C. B., Computational Gasdynamics (1998), Cambridge University Press · Zbl 0947.76001
[41] Levy, D.; Puppo, G.; Russo, G., Compact central WENO schemes for multidimensional conservation laws, SIAM J. Sci. Comput., 22, 2, 656-672 (2000) · Zbl 0967.65089
[42] Chandrasekhar, S., An Introduction to the Study of Stellar Structure (1967), Dover: Dover New York · Zbl 0079.23901
[43] Timmes, F. X.; Swesty, F. D., The accuracy, consistency, and speed of an electron-positron equation of state based on table interpolation of the Helmholtz free energy, Astrophys. J. Suppl. Ser., 126, 2, 501-516 (2000)
[44] Timmes, F. X. (2013), [link]
[45] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes in FORTRAN; The Art of Scientific Computing (1993), Cambridge University Press: Cambridge University Press New York, NY, USA
[46] Gabriel, E.; Fagg, G. E.; Bosilca, G.; Angskun, T.; Dongarra, J. J.; Squyres, J. M.; Sahay, V.; Kambadur, P.; Barrett, B.; Lumsdaine, A.; Castain, R. H.; Daniel, D. J.; Graham, R. L.; Woodall, T. S., Open MPI: goals, concept, and design of a next generation MPI implementation, (Proceedings, 11th European PVM/MPI Users’ Group Meeting. Proceedings, 11th European PVM/MPI Users’ Group Meeting, Budapest, Hungary (2004)), 97-104
[47] Hierarchical data format, version 5, (1997-NNNN)
[48] Jones, E.; Oliphant, T.; Peterson, P., SciPy: open source scientific tools for Python (2001)
[49] Hunter, J. D., Matplotlib: a 2d graphics environment, Comput. Sci. Eng., 9, 3, 90-95 (2007)
[50] Collette, A., Python and HDF5 (2013), O’Reilly
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.