Marshak Boundary Research Articles

Further Development of the Elliptic PDE Formulation of the P N Approximation and its Marshak Boundary Conditions

The expansion of the radiative transfer equation (RTE) into spherical harmonics results in the P N approximation, consisting of (N + 1)2 simultaneous, first-order partial differential equations (PDEs). This system of equations is generally solved subject to a set of so-called Marshak's boundary conditions, although some ambiguity exists in multidimensional media, for which the set provides more than the necessary number of conditions. In recent work Modest has shown that the general 3-D P N approximation can be formulated as a set of N(N + 1)/2 second-order, elliptic PDEs, using the original set of Marshak's conditions, and which can be solved with standard PDE solution packages. In this article the Marshak boundary conditions are reexamined in the light of the elliptic formulation, culminating in a self-consistent set of N(N + 1)/2 conditions along the boundary of the enclosure. The elliptic set of PDEs is reformulated and reduced considerably by limiting considerations to isotropic scattering. As an example, the 2-D P 3 approximation is extracted, and sample 2-D P 1, P 3, and P 5 computations are compared with Monte Carlo results.

Analytical solution of P3 approximation to radiative transfer equation for an infinite homogenous media and its validity

The P3 approximation of the spherical harmonics method for the radiative transfer equation whose homogenous solution is the spherical Bessel function is a nonlinear partial differential equation system. After breaking the spherical Bessel function into two exponential terms, we obtain its particular solution by the modified method of variation of parameters. The complete solution of this approximation is not uniquely determined by the Marshak or Mark boundary condition on the boundary (r = 0). Therefore, the constants of the full solution are determined by direct application of the energy conservation, and by the fact that the P3 approximation equals the P1 approximation when the absorption coefficient is much lower than the reduced scattering coefficient. With the analytic solution of the P3 approximation, the spherical harmonics method for the radiative transfer equation can predict accurately the optical property of a large absorption tissue with the ratio of the reduced scattering coefficient to absorption coefficient ranging over 3–10.

Computational techniques for radiative transfer by spherical harmonics

Spherical harmonics may be used to solve radiative transfer problems with extreme forward scattering, provided that the P N approximation is truncated with a quantity that depends on the angle of incidence, and harmonics of high degree are summed separately [W. H. Wells and J. J. Sidorowich, Ann. Phys. 144, 203 (1982)]. In slab geometry this method is compatible with Marshak's boundary conditions and with an extra integration using a formal solution. Acceptable accuracy is obtainable using approximations as low as P 3− P 7. Analytic schemes improve the convergence of high harmonics and assist in selection of a P N m approximation that matches the angle of incidence for best accuracy. As computational aids, we tabulate roots of P n m and useful sums and integrals that may be regarded as Legendre transforms and their inverse.

Preliminary Calculation for Experiments Using Osaka University HITS, (I)

In order to discuss the ambiguity inherent in the diffusion approximation applied to a small-sized medium including a pulsed neutron source, the time variation and the spatial distribution of thermal neutron angular flux are calculated in P 3-approximation of the time-dependent Boltzmann transport equation. With the discussion on the existence of a set of real and complex eigenvalues, the real eigenvalues for the lowest and some higher modes are computed numerically and the corresponding eigenfunctions are derived under Marshak's boundary condition. The results are compared with those obtained in the diffusion approximation and the physical interpretations of resultant asymptotic and non-asymptotic thermal neutron flux are given.