High-Order Accurate Structure-Preserving Finite Volume Scheme for Ten-Moment Gaussian Closure Equations with Source Terms: Positivity and Well-Balancedness

References

[1] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. 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. 10.1137/S1064827503431090Search in Google Scholar

[2] C. Berthon, Numerical approximations of the 10-moment Gaussian closure, Math. Comp. 75 (2006), no. 256, 1809–1831. 10.1090/S0025-5718-06-01860-6Search in Google Scholar

[3] C. Berthon, B. Dubroca and A. Sangam, An entropy preserving relaxation scheme for ten-moments equations with source terms, Commun. Math. Sci. 13 (2015), no. 8, 2119–2154. 10.4310/CMS.2015.v13.n8.a7Search in Google Scholar

[4] B. Biswas, H. Kumar and A. Yadav, Entropy stable discontinuous Galerkin methods for ten-moment Gaussian closure equations, J. Comput. Phys. 431 (2021), Article ID 110148. 10.1016/j.jcp.2021.110148Search in Google Scholar

[5] R. Borges, M. Carmona, B. Costa and W. S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys. 227 (2008), no. 6, 3191–3211. 10.1016/j.jcp.2007.11.038Search in Google Scholar

[6] F. Bouchut and X. Lhébrard, A multi well-balanced scheme for the shallow water MHD system with topography, Numer. Math. 136 (2017), no. 4, 875–905. 10.1007/s00211-017-0865-ySearch in Google Scholar

[7] J. Britton and Y. Xing, Well-balanced discontinuous Galerkin methods for the one-dimensional blood flow through arteries model with man-at-eternal-rest and living-man equilibria, Comput. & Fluids 203 (2020), Article ID 104493. 10.1016/j.compfluid.2020.104493Search in Google Scholar

[8] S. L. Brown, P. L. Roe and C. P. T. Groth, Numerical Solution of a 10-moment model for nonequilibrium gasdynamics, 12th AIAA Computational Fluid Dynamics Conference, AIAA, Reston (1995), 337–351. 10.2514/6.1995-1677Search in Google Scholar

[9] P. Chandrashekar and C. Klingenberg, A second order well-balanced finite volume scheme for Euler equations with gravity, SIAM J. Sci. Comput. 37 (2015), no. 3, B382–B402. 10.1137/140984373Search in Google Scholar

[10] S. Ding and K. Wu, A new discretely divergence-free positivity-preserving high-order finite volume method for ideal MHD equations, SIAM J. Sci. Comput. 46 (2024), no. 1, A50–A79. 10.1137/23M1562081Search in Google Scholar

[11] C. Dong, L. Wang, A. Hakim, A. Bhattacharjee, J. A. Slavin, G. A. DiBraccio and K. Germaschewski, Global ten-moment multifluid simulations of the solar wind interaction with mercury: From the planetary conducting core to the dynamic magnetosphere, Geophys. Res. Lett. 46 (2019), 11584–11596. 10.1029/2019GL083180Search in Google Scholar

[12] J. Du, Y. Yang and F. Zhu, Well-balanced positivity-preserving high-order discontinuous Galerkin methods for Euler equations with gravitation, J. Comput. Phys. 505 (2024), Article ID 112877. 10.1016/j.jcp.2024.112877Search in Google Scholar

[13] B. Dubroca, M. Tchong, P. Charrier, V. Tikhonchuk and J.-P. Morreeuw, Magnetic field generation in plasmas due to anisotropic laser heating, Phys. Plasmas 11 (2004), 3830–3839. 10.1063/1.1760089Search in Google Scholar

[14] N. Firouzi Farrashbandi and M. Eslami-Kalantari, Inverse Bremsstrahlung absorption in laser-fusion plasma, J. Theoret. Appl. Phys. 14 (2020), 261–264. 10.1007/s40094-020-00375-4Search in Google Scholar

[15] E. Franck and L. 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. 10.1007/s10915-016-0199-4Search in Google Scholar

[16] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001), no. 1, 89–112. 10.1137/S003614450036757XSearch in Google Scholar

[17] 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. 10.1137/0733001Search in Google Scholar

[18] 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. 10.1016/j.jcp.2018.11.018Search in Google Scholar

[19] G. Hernandez-Duenas and G. Ramirez-Santiago, A well-balanced positivity-preserving central-upwind scheme for one-dimensional blood flow models, Internat. J. Numer. Methods Fluids 93 (2021), no. 2, 369–395. 10.1002/fld.4887Search in Google Scholar

[20] X. Y. Hu, N. A. Adams and C.-W. Shu, Positivity-preserving method for high-order conservative schemes solving compressible Euler equations, J. Comput. Phys. 242 (2013), 169–180. 10.1016/j.jcp.2013.01.024Search in Google Scholar

[21] H. Jiang, H. Tang and K. Wu, Positivity-preserving well-balanced central discontinuous Galerkin schemes for the Euler equations under gravitational fields, J. Comput. Phys. 463 (2022), Article ID 111297. 10.1016/j.jcp.2022.111297Search in Google Scholar

[22] E. A. Johnson and J. A. Rossmanith, Ten-moment two-fluid plasma model agrees well with PIC/Vlasov in GEM problem, Hyperbolic Problems—Theory, Numerics and Applications. Volume 2, Ser. Contemp. Appl. Math. CAM 18, World Scientific, Singapore (2012), 461–468. 10.1142/9789814417099_0045Search in Google Scholar

[23] F. Kanbar, R. Touma and C. Klingenberg, Well-balanced central scheme for the system of MHD equations with gravitational source term, Commun. Comput. Phys. 32 (2022), no. 3, 878–898. 10.4208/cicp.OA-2022-0067Search in Google Scholar

[24] R. Käppeli and S. Mishra, Well-balanced schemes for the Euler equations with gravitation, J. Comput. Phys. 259 (2014), 199–219. 10.1016/j.jcp.2013.11.028Search in Google Scholar

[25] R. Käppeli and S. Mishra, A well-balanced finite volume scheme for the Euler equations with gravitation-the exact preservation of hydrostatic equilibrium with arbitrary entropy stratification, Astronom. & Astrophys. 587 (2016), Paper No. A 94. 10.1051/0004-6361/201527815Search in Google Scholar

[26] 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. 10.1137/18M1196704Search in Google Scholar

[27] C. D. Levermore and W. J. Morokoff, The Gaussian moment closure for gas dynamics, SIAM J. Appl. Math. 59 (1999), no. 1, 72–96. 10.1137/S0036139996299236Search in Google Scholar

[28] G. Li and Y. Xing, High order finite volume WENO schemes for the Euler equations under gravitational fields, J. Comput. Phys. 316 (2016), 145–163. 10.1016/j.jcp.2016.04.015Search in Google Scholar

[29] G. Li and Y. Xing, Well-balanced discontinuous Galerkin methods for the Euler equations under gravitational fields, J. Sci. Comput. 67 (2016), no. 2, 493–513. 10.1007/s10915-015-0093-5Search in Google Scholar

[30] A. K. Meena and H. Kumar, Robust MUSCL schemes for ten-moment Gaussian closure equations with source terms, Int. J. Finite Vol. 14 (2017), 1–34. Search in Google Scholar

[31] A. K. Meena and H. Kumar, A well-balanced scheme for ten-moment Gaussian closure equations with source term, Z. Angew. Math. Phys. 69 (2018), no. 1, Paper No. 8. 10.1007/s00033-017-0901-xSearch in Google Scholar

[32] A. K. Meena and H. Kumar, Robust numerical schemes for two-fluid ten-moment plasma flow equations, Z. Angew. Math. Phys. 70 (2019), no. 1, Paper No. 23. 10.1007/s00033-018-1061-3Search in Google Scholar

[33] A. K. Meena, H. Kumar and P. Chandrashekar, Positivity-preserving high-order discontinuous Galerkin schemes for ten-moment Gaussian closure equations, J. Comput. Phys. 339 (2017), 370–395. 10.1016/j.jcp.2017.03.024Search in Google Scholar

[34] A. K. Meena, R. Kumar and P. Chandrashekar, Positivity-preserving finite difference WENO scheme for ten-moment equations with source term, J. Sci. Comput. 82 (2020), no. 1, Paper No. 15. 10.1007/s10915-019-01110-1Search in Google Scholar

[35] J.-P. Morreeuw, A. Sangam, B. Dubroca, P. Charrier and V. Tikhonchuk, Electron temperature anisotropy modeling and its effect on anisotropy-magnetic field coupling in an underdense laser heated plasma, J. Phys. IV 133 (2006), 295–300. 10.1051/jp4:2006133058Search in Google Scholar

[36] E. Pimentel-García, L. O. Müller, E. F. Toro and C. Parés, High-order fully well-balanced numerical methods for one-dimensional blood flow with discontinuous properties, J. Comput. Phys. 475 (2023), Article ID 111869. 10.1016/j.jcp.2022.111869Search in Google Scholar

[37] Y. Ren, K. Wu, J. Qiu and Y. Xing, On high order positivity-preserving well-balanced finite volume methods for the Euler equations with gravitation, J. Comput. Phys. 492 (2023), Article ID 112429. 10.1016/j.jcp.2023.112429Search in Google Scholar

[38] A. Sangam, An HLLC scheme for ten-moments approximation coupled with magnetic field, Int. J. Comput. Sci. Math. 2 (2008), no. 1–2, 73–109. 10.1504/IJCSM.2008.019724Search in Google Scholar

[39] A. Sangam, J.-P. Morreeuw and V. Tikhonchuk, Anisotropic instability in a laser heated plasma, Phys. Plasmas 14 (2007), Article ID 053111. 10.1063/1.2736347Search in Google Scholar

[40] C. Sen and H. Kumar, Entropy stable schemes for ten-moment Gaussian closure equations, J. Sci. Comput. 75 (2018), no. 2, 1128–1155. 10.1007/s10915-017-0579-4Search in Google Scholar

[41] A. Thomann, M. Zenk and C. 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. 10.1002/fld.4703Search in Google Scholar

[42] D. Turnbull, J. Katz, M. Sherlock, L. Divol, N. R. Shaffer, D. J. Strozzi, A. Colaïtis, D. H. Edgell, R. K. Follett and K. R. McMillen, Inverse bremsstrahlung absorption, Phys. Rev. Lett. 130 (2023), Article ID 145103. 10.1103/PhysRevLett.130.145103Search in Google Scholar PubMed

[43] D. Varma and P. Chandrashekar, A second-order, discretely well-balanced finite volume scheme for Euler equations with gravity, Comput. & Fluids 181 (2019), 292–313. 10.1016/j.compfluid.2019.02.003Search in Google Scholar

[44] C. Wang, X. Zhang, C.-W. Shu and J. Ning, Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations, J. Comput. Phys. 231 (2012), no. 2, 653–665. 10.1016/j.jcp.2011.10.002Search in Google Scholar

[45] J. Wang and H. Tang, A second-order direct Eulerian GRP scheme for ten-moment Gaussian closure equations with source terms, J. Comput. Phys. 523 (2025), Article ID 113671. 10.1016/j.jcp.2024.113671Search in Google Scholar

[46] J. Wang, H. Tang and K. Wu, High-order accurate positivity-preserving and well-balanced discontinuous Galerkin schemes for ten-moment Gaussian closure equations with source terms, J. Comput. Phys. 519 (2024), Article ID 113451. 10.1016/j.jcp.2024.113451Search in Google Scholar

[47] L. Wang, K. Germaschewski, A. Hakim, C. Dong, J. Raeder and A. Bhattacharjee, Electron physics in 3-D two-fluid 10-moment modeling of Ganymede’s magnetosphere, J. Geophys. Res. Space Phys. 123 (2018), 2815–2830. 10.1002/2017JA024761Search in Google Scholar

[48] K. Wu, Positivity-preserving analysis of numerical schemes for ideal magnetohydrodynamics, SIAM J. Numer. Anal. 56 (2018), no. 4, 2124–2147. 10.1137/18M1168017Search in Google Scholar

[49] K. Wu, Minimum principle on specific entropy and high-order accurate invariant-region-preserving numerical methods for relativistic hydrodynamics, SIAM J. Sci. Comput. 43 (2021), no. 6, B1164–B1197. 10.1137/21M1397994Search in Google Scholar

[50] K. Wu, H. Jiang and C.-W. Shu, Provably positive central discontinuous Galerkin schemes via geometric quasilinearization for ideal MHD equations, SIAM J. Numer. Anal. 61 (2023), no. 1, 250–285. 10.1137/22M1486996Search in Google Scholar

[51] K. Wu and C.-W. Shu, Provably positive high-order schemes for ideal magnetohydrodynamics: Analysis on general meshes, Numer. Math. 142 (2019), no. 4, 995–1047. 10.1007/s00211-019-01042-wSearch in Google Scholar

[52] K. Wu and C.-W. Shu, Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations, Numer. Math. 148 (2021), no. 3, 699–741. 10.1007/s00211-021-01209-4Search in Google Scholar

[53] K. Wu and C.-W. Shu, Geometric quasilinearization framework for analysis and design of bound-preserving schemes, SIAM Rev. 65 (2023), no. 4, 1031–1073. 10.1137/21M1458247Search in Google Scholar

[54] K. Wu and H. Tang, High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics, J. Comput. Phys. 298 (2015), 539–564. 10.1016/j.jcp.2015.06.012Search in Google Scholar

[55] K. Wu and H. Tang, Admissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations, Math. Models Methods Appl. Sci. 27 (2017), no. 10, 1871–1928. 10.1142/S0218202517500348Search in Google Scholar

[56] K. Wu and Y. Xing, Uniformly high-order structure-preserving discontinuous Galerkin methods for Euler equations with gravitation: positivity and well-balancedness, SIAM J. Sci. Comput. 43 (2021), no. 1, A472–A510. 10.1137/20M133782XSearch in Google Scholar

[57] Y. Xing and C.-W. Shu, High order finite difference WENO schemes with the exact conservation property for the shallow water equations, J. Comput. Phys. 208 (2005), no. 1, 206–227. 10.1016/j.jcp.2005.02.006Search in Google Scholar

[58] Y. Xing and C.-W. 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. 10.1007/s10915-012-9585-8Search in Google Scholar

[59] Y. Xing, X. Zhang and C.-W. Shu, Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations, Adv. Water Res. 33 (2010), 1476–1493. 10.1016/j.advwatres.2010.08.005Search in Google Scholar

[60] Z. Xu, Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one-dimensional scalar problem, Math. Comp. 83 (2014), no. 289, 2213–2238. 10.1090/S0025-5718-2013-02788-3Search in Google Scholar

[61] X. Zhang, On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier–Stokes equations, J. Comput. Phys. 328 (2017), 301–343. 10.1016/j.jcp.2016.10.002Search in Google Scholar

[62] X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys. 229 (2010), no. 9, 3091–3120. 10.1016/j.jcp.2009.12.030Search in Google Scholar

[63] X. Zhang and C.-W. Shu, On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys. 229 (2010), no. 23, 8918–8934. 10.1016/j.jcp.2010.08.016Search in Google Scholar

[64] Z. Zhang, J. Duan and H. Tang, High-order accurate well-balanced energy stable adaptive moving mesh finite difference schemes for the shallow water equations with non-flat bottom topography, J. Comput. Phys. 492 (2023), Article ID 112451. 10.1016/j.jcp.2023.112451Search in Google Scholar

[65] Z. Zhang, H. Tang and J. Duan, High-order accurate well-balanced energy stable finite difference schemes for multi-layer shallow water equations on fixed and adaptive moving meshes, J. Comput. Phys. 517 (2024), Article ID 113301. 10.1016/j.jcp.2024.113301Search in Google Scholar

[66] Z. Zhang, H. Tang and K. Wu, High-order accurate structure-preserving finite volume schemes on adaptive moving meshes for shallow water equations: well-balancedness and positivity, J. Comput. Phys. 527 (2025), Article ID 113801. 10.1016/j.jcp.2025.113801Search in Google Scholar