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Journal of Scientific Computing

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01-03-2015

A Frequency-Domain Approach for the \(\hbox {P}_1\) Approximation of Time-Dependent Radiative Transfer

Authors: Mohamed Addam, Abderrahman Bouhamidi, Mohammed Seaid

Published in: Journal of Scientific Computing | Issue 3/2015

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Abstract

We propose a new frequency-domain method to solve the simplified \(\text {P}_1\) approximation of time-dependent radiative transfer equations. The method employs the Fourier transform and consists of two stages. In the first stage the equations are transformed into an elliptic problem for the frequency variables. The numerical solutions of this problem are approximated using a Galerkin projection method based on the tensor-product B-spline interpolants. In the second stage a Gauss–Hermite quadrature procedure is proposed for the computation of the inverse Fourier transform to recover the numerical solutions of the original simplified \(\text {P}_1\) problem. The method avoids the discretization of the time variable in the considered system and it accurately resolves all time scales in radiative transfer regimes. Several test examples are used to verify high accuracy, effectiveness and good resolution properties for smooth and discontinuous solutions.

next article Convergence Analysis of a Minimax Method for Finding Multiple Solutions of Semilinear Elliptic Equation: Part I—On Polyhedral Domain

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Abramowitz, M., Setgun, I.A. (eds.): Handbook of mathematical functions with formulas, graphs, and mathematical tables, 9th printing. Dover, New York (1972)

Addam, M.: Approximation du problème de diffusion en tomographie optique et problème inverse. Ph.D. thesis, LMPA, Université Lille-Nord de France (2010)

Addam, M., Bouhamidi, A., Jbilou, K.: Signal reconstruction for the diffusion transport equation using tensorial spline Galerkin approximation. Appl. Numer. Math. 62, 1089–1108 (2012)

Anile, A.M., Pennisi, S., Sammartino, M.A.: A thermodynamical approach to Eddington factors. J. Math. Phys. 32, 544–550 (1991)CrossRefMATHMathSciNet

Backofen, R., Bilz, T., Ribalta, A., Voigt, A.: \({SP}_{N}\)-approximations of internal radiation in crystal growth of optical materials. J. Cryst. Growth. 266, 264–270 (2004)CrossRef

Brunner, T.A., Holloway, J.P.: Two-dimensional time dependent Riemann solvers for neutron transport. J. Comput. Phys. 210, 386–399 (2005)

Campbell, G.A., Foster, R.M.: Fourier Integrals for Practical Applications. D. Van Nostrand Company, New York (1948)

Fiveland, W.: The selection of discrete ordinate quadrature sets for anisotropic scattering. ASME HTD. Fundam. Radiat. Heat Transf. 160, 89–96 (1991)

Frank, M., Klar, A., Larsen, E.W., Yasuda, S.: Time-dependent simplified \({P}_{N}\) approximation to the equations of radiative transfer. J. Comput. Phys. 226, 2289–2305 (2007)CrossRefMATHMathSciNet

10.

Frank, M., Lang, L., Schäfer, M.: Adaptive finite element simulation of the time-dependent simplified \({P}_{N}\) equations. J. Sci. Comput. 226, 2289–2305 (2007)MATH

11.

Gelbard, E.M.: Simplified spherical harmonics equations and their use in shielding problems. Technical Report WAPD-T-1182, Bettis Atomic Power Laboratory (1961)

12.

Gorpas, D., Yova, D., Politopoulos, K.: A three-dimensional finite elements approach for the coupled radiative transfer equation and diffusion approximation modeling in fluorescence imaging. J. Quant. Spectrosc. Radiat. Transf. 111, 553–568 (2010)CrossRef

13.

Klose, A.D., Larsen, E.W.: Light transport in biological tissue based on the simplified spherical harmonics equations. J. Comput. Phys. 220, 441–470 (2006)CrossRefMATHMathSciNet

14.

Klose, A.D., Poschinger, T.: Excitation-resolved fluorescence tomography with simplified spherical harmonics equations. Phys. Med. Biol. 56, 1443–1469 (2011)CrossRef

15.

Kotiluoto, P.: Adaptive tree multigrids and simplified spherical harmonics approximation in deterministic neutral and charged particle transport. Ph.D. thesis, University of Helsinki, VTT Technical Research Centre of Finland (2007)

16.

Larsen, E., Morel, J., McGhee, J.: Asymptotic derivation of the multigroup \({P}_1\) and simplified \({P}_{N}\) equations with anisotropic scattering. Nucl. Sci. Eng. 123, 328–367 (1996)

17.

Larsen, E., Thömmes, G., Klar, A., Seaid, M., Götz, T.: Simplified \({P}_{N}\) approximations to the equations of radiative heat transfer and applications. J. Comput. Phys. 183, 652–675 (2002)CrossRefMATHMathSciNet

18.

Larsen, E.W.: Diffusion theory as an asymptotic limit of transport theory for nearly critical systems with small mean free paths. Ann. Nucl. Energy 7, 249–255 (1980)CrossRef

19.

Lewis, E., Miller, W.: Computational Methods of Neutron Transport. Wiley, New York (1984)

20.

Marshak, R.E.: Note on the spherical harmonic method as applied to the milne problem for a sphere. Phys. Rev. 71, 443–446 (1947)CrossRefMATHMathSciNet

21.

Mihalas, D., Mihalas, B.S.: Foundations of Radiation Hydrodynamics. Oxford University Press, New York (1983)

22.

Modest, M.F.: Radiative Heat Transfer. McGraw-Hill, New York (1993)

23.

Pomraning, G.C.: The Equations of Radiation Hydrodynamics. Pregamon press, Oxford (1973)

24.

Sanz-Serna, J.M., Stuart, A.M.: A note on uniform in time error estimates for approximations to reaction-diffusion equations. IMA J. Numer. Anal. 12, 457–462 (1992)CrossRefMATHMathSciNet

25.

Seaid, M., Klar, A.: Efficient preconditioning of linear systems arising from the discretization of radiative transfer equation. Lect. Notes. Comput. Sci. 35, 211–236 (2003)CrossRefMathSciNet

26.

Su, B., Olson, G.L.: An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium. Ann. Nucl. Energy 24, 1035–1055 (1997)CrossRef

27.

Tarvainen, T., Kolehmainen, V., Arridge, S.R., Kaipio, J.P.: Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model. J. Quant. Spectrosc. Radiat. Transf. 112, 2600–2668 (2011)CrossRef

28.

Teleaga, I., Seaid, M.: Simplified radiative models for low Mach number reactive flows. Appl. Math. Model. 32, 971–991 (2009)

29.

Thömmes, G., Pinnau, R., Seaid, M., Götz, T., Klar, A.: Numerical methods and optimal control for glass cooling processes. Transp. Theory Stat. Phys. 31, 513–529 (2002)CrossRefMATH

30.

Viskanta, R., Anderson, E.E.: Heat transfer in semitransparent solids. Adv. Heat Transf. 11, 317–441 (1975)CrossRef

Title: A Frequency-Domain Approach for the Approximation of Time-Dependent Radiative Transfer
Authors: Mohamed Addam
Abderrahman Bouhamidi
Mohammed Seaid
Publication date: 01-03-2015
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2015
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-014-9870-9

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