Why are the P{sub n} and S{sub n} Methods Equivalent

Cullen, D E

Lawrence Livermore National Laboratory

关键词: 99 General And Miscellaneous//Mathematics, Computing, And Information Science;    Boltzmann Equation;    Discrete Ordinate Method;    Physics;    Legendre Polynomials;   

DOI  :  10.2172/15006206
RP-ID  :  UCRL-ID-145518
RP-ID  :  W-7405-ENG-48
RP-ID  :  15006206

美国|英语

来源: UNT Digital Library

【 摘 要 】

The author assumes that the reader is familiar with the Spherical harmonics, Pn, method and the discrete ordinates, S{sub n}, method; for a derivation of the equations used in these methods. I will only discuss the Boltzmann equation in one dimension, and the Sn method using Gaussian quadrature. I will do this merely to simplify the following discussion; once you understand the concepts presented here you can easily extend the conclusions to more general situations. Why are the spherical harmonics P{sub n} and discrete ordinate S{sub n} methods, or more correctly the P{sub n} and S{sub n+1} methods, equivalent, e.g., P{sub 3} is equivalent to S{sub 4}? When the S{sub n} method uses a Gaussian quadrature most textbooks will tell you that both methods are equivalent to assuming that the angular flux can be represented by a Legendre polynomial expansion of order n. Most textbooks are wrong [1]. We know that the S{sub n} method constrains the ''particles'' to travel in discrete directions; when Gaussian quadrature is used these discrete directions correspond to the zeros of the Legendre polynomial P{sub n+1}({mu}). What is not immediately obvious is that the P{sub n} method constrains the ''particles'' in exactly the same way. That is why the two methods are equivalent. The author discusses this in terms of physics and mathematics.