A New Proof of the Asymptotic Diffusion Limit of the S_N Neutron Transport Equation

Figures & data

TABLE I Eigenvalues of the S12 Gauss Legendre Quadrature*

$\epsilon$	$\sqrt{3{\mathrm{\Sigma }}_t{\mathrm{\Sigma }}_a},$Eq. (42)	$\lambda$, Eq. (37)	Eigenvalues of ${\mathrm{\Sigma }}_t{\mu }^{-1}\left(\mathbf{I}-\frac{c}{2}\mathbf{W}\right)$
			${\lambda }_1$	${\lambda }_2$	${\lambda }_3$	${\lambda }_4$	${\lambda }_5$	${\lambda }_6$
0.01	1.549193	1.549143	1.549143	103.863341	118.318367	151.916040	239.421708	698.797818
0.001	1.549193	1.549192	1.549192	1038.632370	1183.1795710	1519.150373	2394.194436	6987.900279
0.0001	1.549193	1.549193	1.549193	10386.323596	11831.7953000	15191.502734	23941.942097	69878.995006

*${\mathrm{\Sigma }}_t=\frac{1}{\epsilon }$; ${\mathrm{\Sigma }}_a=0.8\epsilon .$

TABLE II Comparison of Diffusion BCs*

$\epsilon$	SN, Eq. (69)				Habetler and Matkowsky, Eq. (67a)
	S16		S512
	${\psi }_i=1$ $\left(i=1,..,8\right)$	${\psi }_i={\mu }_i$ $\left(i=1,..,8\right)$	${\psi }_i=1$ $\left(i=1,..,256\right)$	${\psi }_i={\mu }_i$ $\left(i=1,..,256\right)$	$f\left(\mu \right)=1$	$f\left(\mu \right)=\mu$
0.1	1.792	1.274	1.792	1.276	1.999	1.420
0.01	1.978	1.404	1.978	1.405
0.001	1.998	1.418	1.998	1.419

*${\mathrm{\Sigma }}_t=\frac{1}{\epsilon };{\mathrm{\Sigma }}_a=0.8\epsilon$

Fig. 1. Analytical solution vs. diffusion solution