Discrete Eigenvalues of Case Spectrum with Anisotropic Scattering

On the spectrum of discrete-ordinates neutron transport problems

“…For high quadrature orders, the dominant eigenvalues obtained from the problem should agree with the discrete eigenvalues from the one-speed analytic problem. Sahni and Tureci [4] reported the discrete eigenvalues calculated considering several combinations of values of and for three test cases: (a) linearly anisotropic scattering ( 0 , 1 ≠ 0 , otherwise = 0), (b) isotropic+pure quadratic scattering ( 0 , 2 ≠ 0 , otherwise = 0), and (c) isotropic+pure triplet scattering ( 0 , 3 ≠ 0 , otherwise = 0).…”

“…The eigenvalue problem from the BTE, Equation ( 9), was solved for all the examples reported by Sahni and Tureci [4]. It was started at low orders of the Gauss-Legendre quadrature that were increased up to obtain results in agreement with the discrete Case's spectrum within a range of less than 100 .…”

“…According to the notation used by Sahni and Tureci [4], the steady-state BTE for one-speed neutrons in a slab-geometry homogeneous media can be written as…”

“…Over the past 60 years, the method has been applied to more complex problems that may consider energy dependence, multiplying media, anisotropic scattering, heterogeneity, and/or multiple dimensions [2,4].…”

“…In regard to obtaining a discrete eigenvalue spectrum, besides the monoenergetic problem with isotropic scattering [1], the main results have been applied to problems considering linearly anisotropic scattering [6], and more recently, arbitrary-order anisotropic scattering [4,7,8]. In Section 2 of this paper, the methodology described by Sahni and Tureci [4] is summarized and the main results for all the mentioned cases are presented.…”