The minimum entropy approximation to the radiative transfer equation. (English) Zbl 1186.85023
Tadmor, Eitan (ed.) et al., Hyperbolic problems. Theory, numerics and applications. Contributed talks. Proceedings of the 12th international conference on hyperbolic problems, June 9–13, 2008. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4730-5/hbk; 978-0-8218-4728-2/2-vol. set). Proceedings of Symposia in Applied Mathematics 67, Part 2, 987-996 (2009).
To ensure the optimal treatment of cancer using radiotherapy, the administered dose must be accurate to within a few percent. However the models used in clinical dose calculation today fail to achieve this accuracy in the inhomogeneities encountered in tissues and organs such as bone, lungs and sinus passages. With the aim of improving upon the currently available heuristic models, a deterministic method is considered: The radiative transfer equation is solved using the method of moments. The closure of the thereat derived system of moment equations is obtained using the minimum entropy principle.
The present article summarizes the results of former works, in which it was shown, that the minimum entropy closure ensures the hyperbolicity of the moment system and non-negativity of the distribution function [A. M. Anile, S. Pennisi and M. Sammartino, J. Math. Phys. 32, No. 2, 544–550 (1991; Zbl 0850.76881); B. Dubroca and J.-L. Feugeas, C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 10, 915–920 (1999; Zbl 0940.65157)]. The first order minimum entropy system, however, produces unphysical shocks in numerical experiments involving two photon beams [Brunner and Holloway, J. Quant. Spectrosc. Radiat. Transfer 69, 543–566 (2001); M. Frank, H. Hensel and A. Klar, SIAM J. Appl. Math. 67, No. 2, 582–603 (2007; Zbl 1123.78011)].
In the present work, the second order moment system is closed by numerically inverting the resulting non-linear system. By avoiding the use of iterative procedures, it is ensured that the closure is calculated to extremely high precision. The properties of the resulting system and the highest order moment \(N_3(N_1,N_2)\) are investigated. It is shown that the underlying distribution function on the boundary of the moments’ admissible domain becomes a linear combination of two Dirac delta functions, a fact that bodes well for overcoming the shortcomings of the first order minimum entropy system.
For the entire collection see [Zbl 1179.35009].
The present article summarizes the results of former works, in which it was shown, that the minimum entropy closure ensures the hyperbolicity of the moment system and non-negativity of the distribution function [A. M. Anile, S. Pennisi and M. Sammartino, J. Math. Phys. 32, No. 2, 544–550 (1991; Zbl 0850.76881); B. Dubroca and J.-L. Feugeas, C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 10, 915–920 (1999; Zbl 0940.65157)]. The first order minimum entropy system, however, produces unphysical shocks in numerical experiments involving two photon beams [Brunner and Holloway, J. Quant. Spectrosc. Radiat. Transfer 69, 543–566 (2001); M. Frank, H. Hensel and A. Klar, SIAM J. Appl. Math. 67, No. 2, 582–603 (2007; Zbl 1123.78011)].
In the present work, the second order moment system is closed by numerically inverting the resulting non-linear system. By avoiding the use of iterative procedures, it is ensured that the closure is calculated to extremely high precision. The properties of the resulting system and the highest order moment \(N_3(N_1,N_2)\) are investigated. It is shown that the underlying distribution function on the boundary of the moments’ admissible domain becomes a linear combination of two Dirac delta functions, a fact that bodes well for overcoming the shortcomings of the first order minimum entropy system.
For the entire collection see [Zbl 1179.35009].
Reviewer: Claudia-Veronika Meister (Darmstadt)
MSC:
| 85A25 | Radiative transfer in astronomy and astrophysics |
| 78M05 | Method of moments applied to problems in optics and electromagnetic theory |
