Diffusion Methods

Methods for solving the kinetic equations of neutron (or other particle) transport, modifying the equations of the diffusion approximation. Since a diffusion approximation yields the correct form of the asymptotic solution of the transport equation (far away from the sources and from interfaces between media with differing properties), its improvements consist in a correct choice of the constants (e.g. the diffusion coefficient) and an intelligent formulation of boundary conditions with the vacuum and between domains with differing physical characteristics.

The improved diffusion method utilizes, in the single-velocity problem, a transcendental equation for an infinite medium,

$$ p \frac{ { \mathop{\rm Ar} \mathop{\rm tanh} } k }{k} = 1 , $$

to determine the diffusion coefficient

$$ D _ {0} = ( 1- \frac{p)}{k ^ {2} } , $$

where $ p $ is the ratio of the dispersion section to the complete section and $ k $ is the root of the characteristic equation. At medium interfaces, at extrapolated points, boundary conditions are imposed; these are obtained from the exact solution of a two-media problem with a constant complete section (equality of logarithmic derivatives and jumps of asymptotic densities).

Another way of improving the diffusion approximation is the use of the $ P _ {2} $- approximation of the method of spherical harmonics (cf. Spherical harmonics, method of). The ordinary diffusion approximation starts out from a $ P _ {1} $- approximation of the method of spherical harmonics. The passage to the $ P _ {2} $- approximation yields a diffusion equation with corrected parameters and refined boundary conditions, the neutron density on the interface being discontinuous.

The solution of the diffusion equation may also be employed to accelerate the convergence of successive approximations of the kinetic transport equation, involving in the next iteration the use of the approximate solution of the kinetic equation for the calculation of corrections to the diffusion coefficient.

It is also possible to combine, in the framework of a single problem, the diffusion solution with the exact solution, in which the diffusion approximation is employed away from domains occupied by absorbers, sources, etc., while solving the exact transport equation in such domains.

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See also Diffusion approximation; Diffusion equation.

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