A Spherical-harmonics Method for Multi-group or Non-gray Radiation Transport

The spherical-harmonics method, also called the PN method, is used to develop solutions to a class of multi-group or non-gray radiation transport problems. The multigroup model considered allows an anisotropic scattering law and transfer from any group to any group. In addition to a spherical-harmonics solution for the case of a homogeneous radiative-transfer equation, a particular solution for the PN method is derived for the case of multi-group radiative transfer in a homogeneous plane-parallel medium that contains group sources that vary with position and direction. Computational aspects of the developed solutions are discussed, and numerical results for a test case are reported. INTRODUCTION We consider here the multi-group or non-gray radiation transport equation written as P; ‘P(G P) + S’p(z, CL) =; i P,(P)T, I-O s I J',W)~P(z, P') b + W P) (1) -I for z E (0, z,,) and p E [ 1, 11. Here the Legendre polynomials are denoted by P,&), and the transfer matrices T, are such that particle transfer (by, say, scattering and/or fission) between and within all energy groups is allowed. In addition, the elements tj, (z, p), &(z, p), . . . , t,b,,,(z, p) of the M-vector Y(z, CL) are the group angular fluxes or intensities, the elements s, , s,, . . . , sy of the diagonal S matrix are the group total cross sections, z is the position variable measured in cm and p is the direction cosine, with respect to the positive z axis, that defines the direction of motion. Finally, we use E(z, ,u) in Eq. (1) to represent an inhomogeneous (specified) source that could describe, for example, spontaneous emission or could be present in the equation because of a mathematical decomposition, as Chandrasekhar’ did, of some previously formulated problem. Along with Eq. (l), we consider here boundary conditions of the form Y’(O, CL) = F,(P) (2a) and ‘p(% -P) = F&) (2b) for p E [0, 11. Here F,(p) and F,(p) are considered given. In order to use dimensionless units we introduce an optical variable T = ZS,in and an optical thickness TV = ZgS,in, where s,,,~,, is the minimum of the set {si}, and rewrite Eqs. (1) and (2) as for z e(O,t,) and /J E[-1, 11, and y(O, P) = F, (cl) (da) and P(%> -P) = F,(P), (4b) for p E [0, 11. Here the diagonal matrix Z has entries ci = Si/S,in, the dimensionless transfer matrices are defined by C, = T,/s,i” and Q(r, p) = E(z, p)/S,i,. 95 96 C. E. SIEWERT THE HOMOGENEOUS EQUATION Following our previous work, for example Refs. 2-5, with the spherical-harmonics method, we express our approximate PN solution to the homogeneous version of Eq. (3) as N 21+1 Ypc(r, P) = c I=0 2 P,(p) i [A,em”5j + (l)‘B, e.~(‘O~‘)‘5,]G,(~,)N(5j) ,=I where the constants Aj and B, are to be fixed by the boundary conditions. We consider N to be odd, and so the spectrum is given by 5 = <,, j = 1,2, . . . , J = M(N + 1)/2. Here the A4 x Mmatrix of polynomials G,(t) are the result of a multi-group extension of the Chandrasekhar polynomials,’ and the lj denote the J zeros of det G, + , (4) that lie in the right half-plane. Finally the vector N(t,) is used to denote a null-vector of G,, , (r,). To be specific, we note from Ref. 6 that the matrix version of the Chandrasekhar polynomials required here can be defined by the starting value Go(t)=1 (6) and the three-term recursion formula t-W,(l) = (I+ l>G+, (4) + IG, , (5) (7) forl=O,l,.... Here I denotes the M x A4 identity matrix and forI=O,l,...,Land h,=(21+ l)Z -C,