The Discrete Ordinates Method For Solving The Transport Equation With Anisotropic Scattering

Anisotropic scattering is an important research area in reactor physics, astronomical physics and radiation transfer. For the complexity of transport equation, it's hard to be analytic solved except for special conditions. Discrete ordinate method is a simple process and widely applied in numerical study of transport in practical problems.In the Discrete Ordinate Method (DOM) the transport equation is solved for a set of m discrete directions. Each direction Ω_m is associated with a solid angle in which the intensity is assumed to be constant. All solid angles are non-overlapping and spanning the total angle range of 4π. The integrals over the direction are replaced by numerical quadrature weights w_m assumedover each ordinate. In each direction the transport equation may be solved analytically (i.e. Without truncation errors due to spatial differencing), and the solutions are exact in the direction Ω_m . The ray effect anomalies arise, however, not from the inability to calculate the angular flux in the discrete ordinate direction exactly. Rather the appear from the inability of the quadrature formulaIn 1 -D plane geometry, spherical geometry or finite cylinder geometry including anisotropic scattering, physical analysis and numerical results show that the distribution function of angular flux is smooth and continuous, with maximum in one or two directions. The higher was the order of anisotropic scattering, the more contribution the angular flux in the directions of peak done. Making good use of angular flux in different directions, structuring the high order part of the angular flux in whole to revise the errors in quadrature, we can improve the calculating accuracy on the whole.For the anisotropic scattering problem in 1-D slab, according to the special solution of delta function, we split the angular flux function into a finite width delta function (high order) and a remained function. Then the high order function is integrated and the lower order function quadrature. Numerical results show that we can get satisfied accuracy with less discrete directions.This paper has six chapters. After simply introduced the essential concepts of transport theory, we give the differential transport equation and integral equation, and derivate the expressions of transfer operators in different orthogonal ordinates. Then we show several solution methods of transport equation. In Chap 4, we analyze the accuracy of different spatial and angular differial form and the construction of quadrature formula. After taking high accuracy differential forms and quadrature formula, numerical results show that the computing accuracy of isotropic or low order anisotropic scattering has some progress, however, it makes poor improvement in high order amsotropic scattering problem. In Chap 5, we discuss the concepts and applications of analysis discrete ordinate methods by 1-D slab with anisotropic scattering. In the end, the next step is directed.