Spectral Green's function method for neutron transport: isotropic, forward, and backward scattering in 1-D slab geometry

S4 solution of the transport equation for eigenvalues using Legendre polynomials

“…The transport equation in onedimensional geometry is first written in the form of S N by using the integral transform technique with the evenorder Gauss-Legendre quadrature set. A dispersion relation is obtained by using the reasonable homogeneous solution in the S N form of the transport [4,5]. This order of approximation can be spelled to as unnecessary for the calculations in slab geometry.…”

Numerical results for the transport equation with strongly anisotropic scattering in a slab

“…(6) depends on the critical r 0 , the eigenvalue problem 1 max jqj : q 2 specL g ðr 0 Þ È É À r 0 ¼ 0 ð38Þ Table 6 Comparison between the values calculated for critical thickness (L) when k 0 ¼ 1; a 0 ¼ b 0 ¼ 0:05; x ¼ 1 þ b1ll 0 and, length of the slab given by Sahni et al (1992) Table 9 Numerical results for total intensity J 0 ðyÞ, when r 0 ¼ 1:0; a 0 ¼ b 0 ¼ 0:05; L ¼ 1; B0 ¼ BL ¼ 1:0; Q ðyÞ ¼ e Ày and x ¼ 1 þ b1ll 0 þ b2P2ðlÞP2ðl 0 Þ þ b3P3ðlÞP3ðl 0 Þ for some values de b0; b1; b2; b3 and y using N ¼ 400. Table 8 Numerical results for critical thickness when k 0 ¼ 1; r 0 ¼ 1:5; a 0 ¼ b 0 ¼ 0:05 and Table 7 Comparison between the values calculated for the total intensity J 1 ðyÞ when L ¼ 10; k 0 ¼ 1; r 0 ¼ 0:99; Q 0; B0 ¼ 1; BL ¼ 0; a 0 ¼ 0:4 and b 0 ¼ 0:4 with the results published by Anli (2001). with specL g ðr 0 Þ denoting the spectrum of L g , is nonlinear and we solve this by the Secant method.…”

“…We observe that there are few reported results for this quantity in the literature. The results of Anli (2001) are for the particular case with L ¼ 10 and Q 0 and the Gauss-Legendre quadrature with only four ordinates applied in that work does not ensure great numerical precision. Table 7 compare our results for k 0 ¼ 1; r 0 ¼ 0:99; B 0 ¼ 1; B L ¼ 0; a 0 ¼ 0:4 and b 0 ¼ 0:4.…”

“…Ganapol and Kornreich (1996) simulated the scalar flux by their Green's function method. Anli (2001) used a spectral Green's function method and the diamond difference scheme to obtain the total intensity. Awatif and Elghazaly (2004) applied variational techniques to calculate the albedo while Öztürk (2014) used U N method to compute the critical thickness.…”