High-order spherical harmonics&#x2013;nodal collocation scheme for the numerical solution of the time-dependent radiative transfer equation

María T. Capilla; César F. Talavera

doi:10.1364/JOSAA.36.000038

Journal of the Optical Society of America A
Vol. 36,
Issue 1,
pp. 38-50
(2019)
•https://doi.org/10.1364/JOSAA.36.000038

High-order spherical harmonics–nodal collocation scheme for the numerical solution of the time-dependent radiative transfer equation

María T. Capilla and César F. Talavera

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María T. Capilla and César F. Talavera, "High-order spherical harmonics–nodal collocation scheme for the numerical solution of the time-dependent radiative transfer equation," J. Opt. Soc. Am. A 36, 38-50 (2019)

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Abstract

The validity of the spherical harmonics–nodal collocation approximation to the stationary Boltzmann equation has been established in multidimensional neutron transport problems. This is a high-order approximation method that allows a coarse spatial discretization without losing accuracy. We extend the method to solve the time-dependent radiative transfer equation for absorbing media with anisotropic scattering, also incorporating into the model reflecting boundary conditions, due to refractive index mismatching. The formalism is then applied to numerical test cases in one and two spatial dimensions that, using typical values in optical tomography for the physical parameters, check the accuracy and convergence of the method.

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	Top Side ( $x$ axis, $y = 3 cm$ )			Left Side ( $y$ axis, $x = 0 cm$ )
	Source A	Source B	Source C	Source A	Source B	Source C
$μ_{s}^{'}$ ( ${cm}^{- 1}$ )	( $μ_{a} = 0.175 {cm}^{- 1}$ , $g = 0.8$ )
2.9	1.33	2.61	2.91	3.29	6.14	22.69
4.35	0.45	1.65	1.62	0.77	0.69	4.81
5.8	0.85	1.37	1.19	2.43	3.18	3.05
8.7	1.79	1.93	2.21	5.25	7.04	9.21
11.6	2.42	2.68	3.41	7.07	9.16	11.81
$μ_{a}$ ( ${cm}^{- 1}$ )	( $μ_{s} = 29 {cm}^{- 1}$ , $g = 0.8$ )
0.05	0.64	3.32	4.94	7.27	10.33	19.42
0.1	0.71	2.38	3.09	2.81	3.06	6.53
0.175	0.85	1.37	1.19	2.43	3.18	3.05
0.25	1.03	1.13	1.69	5.13	6.65	7.50
0.35	1.32	1.84	3.49	7.63	9.41	10.68
$g$	( $μ_{s} = 29 {cm}^{- 1}$ , $μ_{a} = 0.175 {cm}^{- 1}$ )
0.7	1.79	1.93	2.21	5.27	7.06	9.19
0.75	1.37	1.57	1.58	4.02	5.45	6.86
0.8	0.85	1.37	1.19	2.43	3.18	3.05
0.85	0.45	1.66	1.65	0.80	0.68	4.71
0.9	1.32	2.64	2.98	3.45	6.29	21.98

	Top Side ( $x$ axis, $y = 4 cm$ )			Left Side ( $y$ axis, $x = 0 cm$ )
$μ_{s}$ ( ${cm}^{- 1}$ )	Source A	Source B	Source C	Source A	Source B	Source C
2.5	1.43	1.42	2.06	1.44	4.28	5.08
7.5	1.54	1.56	2.25	1.88	4.83	5.54
12.5	1.61	1.66	2.38	2.21	5.23	5.85
17.5	1.66	1.74	2.49	2.49	5.55	6.10
22.5	1.69	1.79	2.57	2.73	5.82	6.30

	SHNC—Order of Approximations
Cases $μ_{a}$ ( ${mm}^{- 1}$ )	$P_{1}$	$P_{3}$	$P_{7}$
0.0	$1.118 \times 10^{- 3}$	$1.158 \times 10^{- 3}$	$1.165 \times 10^{- 3}$
0.1	$5.126 \times 10^{- 9}$	$7.151 \times 10^{- 9}$	$7.130 \times 10^{- 9}$
0.2	$1.661 \times 10^{- 12}$	$4.388 \times 10^{- 12}$	$4.366 \times 10^{- 12}$

SHNC Order	CPU Times (min)
$P_{1}$	2.20
$P_{3}$	30.83
$P_{7}$	261.94

	Top Side ( $x$ axis, $y = 3 cm$ )			Left Side ( $y$ axis, $x = 0 cm$ )
	Source A	Source B	Source C	Source A	Source B	Source C
$μ_{s}^{'}$ ( ${cm}^{- 1}$ )	( $μ_{a} = 0.175 {cm}^{- 1}$ , $g = 0.8$ )
2.9	1.33	2.61	2.91	3.29	6.14	22.69
4.35	0.45	1.65	1.62	0.77	0.69	4.81
5.8	0.85	1.37	1.19	2.43	3.18	3.05
8.7	1.79	1.93	2.21	5.25	7.04	9.21
11.6	2.42	2.68	3.41	7.07	9.16	11.81
$μ_{a}$ ( ${cm}^{- 1}$ )	( $μ_{s} = 29 {cm}^{- 1}$ , $g = 0.8$ )
0.05	0.64	3.32	4.94	7.27	10.33	19.42
0.1	0.71	2.38	3.09	2.81	3.06	6.53
0.175	0.85	1.37	1.19	2.43	3.18	3.05
0.25	1.03	1.13	1.69	5.13	6.65	7.50
0.35	1.32	1.84	3.49	7.63	9.41	10.68
$g$	( $μ_{s} = 29 {cm}^{- 1}$ , $μ_{a} = 0.175 {cm}^{- 1}$ )
0.7	1.79	1.93	2.21	5.27	7.06	9.19
0.75	1.37	1.57	1.58	4.02	5.45	6.86
0.8	0.85	1.37	1.19	2.43	3.18	3.05
0.85	0.45	1.66	1.65	0.80	0.68	4.71
0.9	1.32	2.64	2.98	3.45	6.29	21.98

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Equations (44)

Journal of the Optical Society of America A