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Some extensions of the $φ$-divergence moment closures for the radiative transfer equation

Published 9 Oct 2023 in math.NA and cs.NA | (2310.05489v1)

Abstract: The $\phi$-divergence-based moment method was recently introduced Abdelmalik et al. (2023) for the discretization of the radiative transfer equation. At the continuous level, this method is very close to the entropy-based MN methods and possesses its main properties, i.e. entropy dissipation, rotational invariance and energy conservation. However, the $\phi$-divergence based moment systems are easier to resolve numerically due to the improved conditioning of the discrete equations. Moreover, exact quadrature rules can be used to compute moments of the distribution function, which enables the preservation of energy conservation, entropy dissipation and rotational invariants, discretely. In this paper we consider different variants of the $\phi$-divergence closures that are based on different approximations of the exponential function and the Planck function. We compare the approximation properties of the proposed closures in the numerical benchmarks.

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References (32)
  1. M. Abdelmalik. Adaptive algorithms for optimal multiscale model hierarchies of the Boltzmann Equation: Galerkin methods for kinetic theory. PhD thesis, TU Eindhoven, 2017.
  2. Moment methods for the radiative transfer equation based on φ𝜑\varphiitalic_φ-divergences. https://arxiv.org/abs/2304.01758 under review, pages 1–29, 2023.
  3. M. Abdelmalik and H. van Brummelen. Moment closure approximations of the boltzmann equation based on φ𝜑\varphiitalic_φ-divergences. J. Stat. Phys., 164:77–104, 2016.
  4. Spectral Methods: Fundamentals in Single Domains. Springer-Verlag, 2006.
  5. B. Carlson. Methods in computational physics. ACADEMIC PRESS, 1963.
  6. S. Chandrasekhar. Radiative transfer. Oxford, 1950.
  7. I. Csiszár. A class of measures of informativity of observation channels. Period. Math. Hung., 2:191–213, 1972.
  8. R. Dautray and J.-L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology: Volume 6, Evolution Problems II. Springer, 2000.
  9. Time-dependent simplified pn approximation to the equations of radiative transfer. J. Comput. Phys., 226(2):2289–2305, 2007.
  10. E. Godlewski and P.-A. Raviart. Numerical approximation of hyperbolic systems of conservation laws. Springer, 1996.
  11. C. D. Hauck. High-order entropy-based closures for linear transport in slab geometry. Commun. Math. Sci, 9(1):187–205, 2011.
  12. Spectral Methods for Time-Dependent Problems. Cambridge, 2009.
  13. K. D. Humbird and R. McClarren. Adjoint-based sensitivity analysis for high-energy density radiative transfer using flux-limited diffusion. High Energy Density Physics, 22:12–16, 2017.
  14. S. Kawashima and W.-A. Yong. Dissipative structure and entropy for hyperbolic systems of balance laws. Arch. Rational Mech. Anal., 174:345–364, 2004.
  15. J.-B. Lasserre. Moment, positive polynomials, and their applications. Imperial college press, 2009.
  16. V.I. Lebedev. Quadratures on a sphere. USSR Comput. Math. Math. Phys., 16(2):10–24, 1976.
  17. A quadrature formula for the sphere of the 131st algebraic order of accuracy. Doklady Mathematics, 59(3):477–481, 1999.
  18. C. D. Levermore. Moment closure hierarchies for kinetic theories. J. Stat. Phys., 83(5–6):1021–1065, 1996.
  19. Computational Methods of Neutron Transport. John Wiley & Sons Inc, 1984.
  20. R. Li and W. Li. 3D B2subscript𝐵2B_{2}italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT model for radiative transfer equation. Int. J. Numer. Anal. Modeling, 17(1):118–150, 2020.
  21. The coupling of radiation and hydrodynamics. The Astro. J., 521:432–450, 1999.
  22. R. McClarren. Theoretical aspects of the simplified pn equations. Transport Theory Stat. Phys., 39(2-4):73–109, 2010.
  23. D. Mihalas and B. R. W. Mihalas. Foundations of Radiation Hydrodynamics. Oxford, 1983.
  24. G. N. Minerbo. Maximum entropy Eddington factors. J. Quant. Spectros. Radiat. Transfer, 20:541–545, 1978.
  25. Diffusion, p1, and other approximate forms of radiation transport. J. Quant. Spec. Rad. Trans., 64(6):619–634, 2000.
  26. T. Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1d case. Kin. rel. models, 13:1243–1280, 2020.
  27. An approximation of the M2subscript𝑀2M_{2}italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT closure: application to radiotherapy dose simulation. J. Sci. Comput., 71:71–108, 2017.
  28. G. C. Pomraning. Equations of Radiation Hydrodynamics. Pergamon, 1973.
  29. A second-order maximum-entropy inspired interpolative closure for radiative heat transfer in gray participating media. J. Quant. Spectros. Radiat. Transfer, page 107238, 2020.
  30. K. Schmuedgen. The moment problem. Springer, 2017.
  31. F. Schneider. Kershaw closures for linear transport equations in slab geometry i: model derivation. J. Comput. Phys., 322:905–919, 2016.
  32. G. Szegö. Orthogonal polynomials. AMS, 1939.

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