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An asymptotic preserving scheme for the \(M_1\) model on polygonal and conical meshes

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Abstract

This work focuses on the design of a 2D numerical scheme for the \(M_1\) model on polygonal and conical meshes. This model is nonlinear and approximates the firsts moments of the radiative transfer equation using an entropic closure. Besides, this model admits a diffusion limit as the cross section goes to infinity. It is important for the numerical scheme to be consistent with this limit, that is to say, it should be asymptotic preserving or AP. Such a scheme already exists on polygonal meshes and the present work consists in adapting it to conical meshes. After introducing conical meshes, we explain the construction of the scheme. It is based on an analogy between the \(M_1\) model and the Euler gas dynamics system. We also present a second order reconstruction procedure and we apply it on both polygonal and conical meshes. In the last section, some numerical test cases are given so as to compare the nodal and conical schemes. The limit scheme is studied and we observe numerically that it is consistent with the diffusion equation.

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References

  1. Mihalas, D., Mihalas, B.W.: Foundations of Radiation Hydrodynamics. Oxford University Press, New York (1984)

    Google Scholar 

  2. Dubroca, B., Feugeas, J.-L.: Étude théorique et numérique d’une hiérarchie de modèles aux moments pour le transfert radiatif. C. R. Acad. Sci. Paris Sér. I Math. 329(10), 915–920 (1999)

    Article  MathSciNet  Google Scholar 

  3. Goudon, T., Lin, C.: Analysis of the m1 model: well-posedness and diffusion asymptotics. J. Math. Anal. Appl. 402(2), 579–593 (2013)

    Article  MathSciNet  Google Scholar 

  4. Bianchini, S., Hanouzet, B., Natalini, R.: Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Commun. Pure Appl. Math. 60(11), 1559–1622 (2007)

    Article  MathSciNet  Google Scholar 

  5. Coulombel, J.-F., Golse, F., Goudon, T.: Diffusion approximation and entropy-based moment closure for kinetic equations. Asymptot. Anal. 45(1–2), 1–39 (2005)

    MathSciNet  Google Scholar 

  6. Goudon, T., Lin, C.: Analysis of the \(M1\) model: well-posedness and diffusion asymptotics. J. Math. Anal. Appl. 402(2), 579–593 (2013)

    Article  MathSciNet  Google Scholar 

  7. Berthon, C., Dubois, J., Turpault, R.: Numerical approximation of the \({\rm M}_1\)-model. In: Mathematical models and numerical methods for radiative transfer, volume 28 of Panoromas Synthèses, pp. 55–86. Soc. Math. France, Paris (2009)

  8. Hauck, C.D., David Levermore, C., Tits, A.L.: Convex duality and entropy-based moment closures: characterizing degenerate densities. SIAM J. Control Optim. 47(4), 1977–2015 (2008)

    Article  MathSciNet  Google Scholar 

  9. Guisset, S., Moreau, J.G., Nuter, R., Brull, S., d’Humières, E., Dubroca, B., Tikhonchuk, V.T.: Limits of the \(M_1\) and \(M_2\) angular moments models for kinetic plasma physics studies. J. Phys. A 48(33), 335501 (2015)

    Article  MathSciNet  Google Scholar 

  10. Jin, S., David Levermore, C.: Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 126(2), 449–467 (1996)

    Article  MathSciNet  Google Scholar 

  11. Buet, C., Despres, B.: Asymptotic preserving and positive schemes for radiation hydrodynamics. J. Comput. Phys. 215(2), 717–740 (2006)

    Article  MathSciNet  Google Scholar 

  12. Buet, C., Cordier, S.: Asymptotic preserving scheme and numerical methods for radiative hydrodynamic models. C. R. Math. Acad. Sci. Paris 338(12), 951–956 (2004)

    Article  MathSciNet  Google Scholar 

  13. Berthon, C., Charrier, P., Dubroca, B.: An HLLC scheme to solve the \(M_1\) model of radiative transfer in two space dimensions. J. Sci. Comput. 31(3), 347–389 (2007)

    Article  MathSciNet  Google Scholar 

  14. Berthon, C., Dubois, J., Dubroca, B., Nguyen-Bui, T.-H., Turpault, R.: A free streaming contact preserving scheme for the \(M_1\) model. Adv. Appl. Math. Mech. 2(3), 259–285 (2010)

    Article  MathSciNet  Google Scholar 

  15. Olbrant, E., Hauck, C.D., Frank, M.: A realizability-preserving discontinuous Galerkin method for the M1 model of radiative transfer. J. Comput. Phys. 231(17), 5612–5639 (2012)

    Article  MathSciNet  Google Scholar 

  16. Buet, C., Després, B., Franck, E.: An asymptotic preserving scheme with the maximum principle for the \(M_1\) model on distorded meshes. C. R. Math. Acad. Sci. Paris 350(11–12), 633–638 (2012)

    Article  MathSciNet  Google Scholar 

  17. Franck, E., Buet, C., Després, B.: Asymptotic preserving finite volumes discretization for non-linear moment model on unstructured meshes. In: Finite Volumes for Complex Applications VI. Problems & Perspectives. Volume 1, 2, volume 4 of Springer Proceedings Mathematics, pp. 467–474. Springer, Heidelberg (2011)

  18. Franck, E.: Construction et analyse numérique de schema asymptotic preserving sur maillages non structurés. Application au transport linéaire et aux systèmes de Friedrichs. PhD thesis, Université Pierre et Marie Curie - Paris VI (2012)

  19. Després, B., Mazeran, C.: Lagrangian gas dynamics in two dimensions and Lagrangian systems. Arch. Ration. Mech. Anal. 178(3), 327–372 (2005)

    Article  MathSciNet  Google Scholar 

  20. Maire, P.H., Abgrall, R., Breil, J., Ovadia, J.: A cell-centered Lagrangian scheme for 2d compressible flow problems. SIAM J. Sci. Comput. 29(4), 1781–1824 (2007)

    Article  MathSciNet  Google Scholar 

  21. Blachère, F., Turpault, R.: An admissibility and asymptotic-preserving scheme for systems of conservation laws with source term on 2D unstructured meshes. J. Comput. Phys. 315, 98–123 (2016)

    Article  MathSciNet  Google Scholar 

  22. Blachère, F., Turpault, R.: An admissibility and asymptotic preserving scheme for systems of conservation laws with source term on 2D unstructured meshes with high-order MOOD reconstruction. Comput. Methods Appl. Mech. Eng. 317, 836–867 (2017)

    Article  MathSciNet  Google Scholar 

  23. Blanc, X., Delmas, V., Hoch, P.: Asymptotic preserving schemes on conical unstructured 2d meshes. Int. J. Numer. Methods Fluids 93(8), 2763–2802 (2021)

    Article  MathSciNet  Google Scholar 

  24. Chidyagwai, P., Frank, M., Schneider, F., Seibold, B.: A comparative study of limiting strategies in discontinuous Galerkin schemes for the \(M_1\) model of radiation transport. J. Comput. Appl. Math. 342, 399–418 (2018)

    Article  MathSciNet  Google Scholar 

  25. Alldredge, G., Schneider, F.: A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension. J. Comput. Phys. 295, 665–684 (2015)

    Article  MathSciNet  Google Scholar 

  26. Kristopher Garrett, C., Hauck, C., Hill, J.: Optimization and large scale computation of an entropy-based moment closure. J. Comput. Phys. 302, 573–590 (2015)

    Article  MathSciNet  Google Scholar 

  27. Guisset, S.: Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier–Stokes equations. Kinet. Relat. Models 13(4), 739–758 (2020)

    Article  MathSciNet  Google Scholar 

  28. Chalons, C., Guisset, S.: An antidiffusive HLL scheme for the electronic \(M_1\) model in the diffusion limit. Multiscale Model. Simul. 16(2), 991–1016 (2018)

    Article  MathSciNet  Google Scholar 

  29. Guisset, S., Brull, S., Dubroca, B., Turpault, R.: An admissible asymptotic-preserving numerical scheme for the electronic \(M_1\) model in the diffusive limit. Commun. Comput. Phys. 24(5), 1326–1354 (2018)

    Article  MathSciNet  Google Scholar 

  30. Guisset, S., Brull, S., D’Humières, E., Dubroca, B.: Asymptotic-preserving well-balanced scheme for the electronic \(M_1\) model in the diffusive limit: particular cases. ESAIM Math. Model. Numer. Anal. 51(5), 1805–1826 (2017)

    Article  MathSciNet  Google Scholar 

  31. Frank, M., Hauck, C.D., Olbrant, E.: Perturbed, entropy-based closure for radiative transfer. Kinet. Relat. Models 6(3), 557–587 (2013)

    Article  MathSciNet  Google Scholar 

  32. Gosse, L., Toscani, G.: An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations. C.R. Math. 334(4), 337–342 (2002)

    Article  MathSciNet  Google Scholar 

  33. Blanc, X., Hoch, P., Lasuen, C.: Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on conical meshes. Working paper or preprint, April (2022)

  34. Bernard-Champmartin, A., Hoch, P., Seguin, N.: Stabilité locale et montée en ordre pour la reconstruction de quantités volumes finis sur maillages coniques non-structurés en dimension 2. preprint https://hal.archives-ouvertes.fr/hal-02497832, March (2020)

  35. Buet, C., Després, B.: A gas dynamics scheme for a two moments model of radiative transfer. Working paper or preprint https://hal.archives-ouvertes.fr/hal-00127189v2, November (2008)

  36. Godlewski, E., Raviart, P.-A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws, volume 118 of Applied Mathematical Sciences, 2nd edn. Springer, New York, NY (2021)

    Book  Google Scholar 

  37. Després, B.: Numerical Methods for Eulerian and Lagrangian Conservation Laws. Birkhäuser, Basel (2017)

    Book  Google Scholar 

  38. Buet, C., Després, B., Franck, E.: Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes. Numer. Math. 122(2), 227–278 (2012)

    Article  MathSciNet  Google Scholar 

  39. Dukowicz, J.K., Kodis, J.W.: Accurate conservative remapping (rezoning) for arbitrary Lagrangian–Eulerian computations. SIAM J. Sci. Stat. Comput. 8(3), 305–321 (1987)

    Article  MathSciNet  Google Scholar 

  40. Hoch, P., Labourasse, E.: A frame invariant and maximum principle enforcing second-order extension for cell-centered ALE schemes based on local convex hull preservation. Int. J. Numer. Methods Fluids 76(12), 1043–1063 (2014)

    Article  MathSciNet  Google Scholar 

  41. Carré, G., Del Pino, S., Després, B., Labourasse, E.: A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension. J. Comput. Phys. 228(14), 5160–5183 (2009)

    Article  MathSciNet  Google Scholar 

  42. Klar, A., Schmeiser, C.: Numerical passage from radiative heat transfer to nonlinear diffusion models. Math. Models Methods Appl. Sci. 11(5), 749–767 (2001)

    Article  MathSciNet  Google Scholar 

  43. Degond, P., Klar, A.: A relaxation approximation for transport equations in the diffusive limit. Appl. Math. Lett. 15(2), 131–135 (2002)

    Article  MathSciNet  Google Scholar 

  44. Wang, L., Yan, B.: An asymptotic-preserving scheme for the kinetic equation with anisotropic scattering: heavy tail equilibrium and degenerate collision frequency. SIAM J. Sci. Comput. 41(1), A422–A451 (2019)

    Article  MathSciNet  Google Scholar 

  45. Jin, S.: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21(2), 441–454 (1999)

    Article  MathSciNet  Google Scholar 

  46. Blankenship, G., Papanicolaou, G.C.: Stability and control of stochastic systems with wide-band noise disturbances. I. SIAM J. Appl. Math. 34(3), 437–476 (1978)

    Article  MathSciNet  Google Scholar 

  47. Bardos, C., Santos, R., Sentis, R.: Diffusion approximation and computation of the critical size. Trans. Am. Math. Soc. 284(2), 617–649 (1984)

    Article  MathSciNet  Google Scholar 

  48. Lions, P.L., Toscani, G.: Diffusive limit for finite velocity Boltzmann kinetic models. Rev. Mat. Iberoam. 13(3), 473–513 (1997)

    Article  MathSciNet  Google Scholar 

  49. Morel, J.E., Densmore, J.D.: A two-component equilibrium-diffusion limit. Ann. Nucl. Energy 32(2), 233–240 (2005)

    Article  Google Scholar 

  50. Larsen, E.W., Keller, J.B.: Asymptotic solution of neutron transport problems for small mean free paths. J. Math. Phys. 15(1), 75–81 (1974)

    Article  MathSciNet  Google Scholar 

  51. Lowrie, R.B., Morel, J.E., Hittinger, J.A.: The coupling of radiation and hydrodynamics*. Astrophys. J. 521(1), 432 (1999)

    Article  Google Scholar 

  52. Morel, J.E., Wareing, T.A., Smith, K.: A linear-discontinuous spatial differencing scheme forsnradiative transfer calculations. J. Comput. Phys. 128(2), 445–462 (1996)

    Article  MathSciNet  Google Scholar 

  53. Larsen, E.W., Pomraning, G.C., Badham, V.C.: Asymptotic analysis of radiative transfer problems. J. Quant. Spectrosc. Radiat. Transf. 29(4), 285–310 (1983)

    Article  Google Scholar 

  54. Blachère, F., Chalons, C., Turpault, R.: Very high-order asymptotic-preserving schemes for hyperbolic systems of conservation laws with parabolic degeneracy on unstructured meshes. Comput. Math. Appl. 87, 41–49 (2021)

    Article  MathSciNet  Google Scholar 

  55. Le Potier, C.: A second order in space combination of methods verifying a maximum principle for the discretization of diffusion operators. C. R. Math. Acad. Sci. Paris 358(1), 89–96 (2020)

    Article  MathSciNet  Google Scholar 

  56. Frankel, J.I., Vick, B., Necati Ozisik, M.: Flux formulation of hyperbolic heat conduction. J. Appl. Phys. 58(9), 3340–3345 (1985)

    Article  Google Scholar 

  57. Molina, J.A.L., Trujillo, M.: Regularity of solutions of the anisotropic hyperbolic heat equation with nonregular heat sources and homogeneous boundary conditions. Turk. J. Math. 41(3), 461–482 (2017)

    Article  MathSciNet  Google Scholar 

  58. Buet, C., Despres, B.: Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics. J. Quant. Spectrosc. Radiat. Transf. 85, 03 (2003)

    Google Scholar 

  59. Boutin, B., Deriaz, E., Hoch, P., Navaro, P.: Extension of ALE methodology to unstructured conical meshes. ESAIM Proc. 32, 31–55 (2011)

    Article  MathSciNet  Google Scholar 

  60. Bernard-Champmartin, A., Deriaz, E., Hoch, P., Samba, G., Schaefer, M.: Extension of centered hydrodynamical schemes to unstructured deforming conical meshes: the case of circles. ESAIM Proc. 38, 135–162 (2012)

    Article  MathSciNet  Google Scholar 

  61. Maire, P.H., Nkonga, B.: Multi-scale Godunov type method for cell-centered discrete Lagrangian hydrodynamics. J. Comput. Phys. 228, 799–821 (2009)

    Article  MathSciNet  Google Scholar 

  62. Roynard, X.: Extension du schéma vofire aux maillages à bords coniques. Technical report, CEA DAM-DIF (2013)

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions (LJLL), Université Paris Cité, CNRS, Sorbonne Université, 75006, Paris, France

    Xavier Blanc

  2. CEA, DAM, DIF, 91297, Arpajon, France

    Philippe Hoch & Clément Lasuen

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  1. Xavier Blanc
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  2. Philippe Hoch
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Blanc, X., Hoch, P. & Lasuen, C. An asymptotic preserving scheme for the \(M_1\) model on polygonal and conical meshes. Calcolo 61, 24 (2024). https://doi.org/10.1007/s10092-024-00574-4

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