Angular Discretization

Discrete Ordinates

The weak form of the neutron transport equation (Governing Equations, Eq. (1)) is discretized in the angular dimension with the use of the discrete ordinates (S_n) method. The NTE is solved for a collection of discrete ordinates (directions) , and a suitable integration method is used to compute the angular integral using the solutions along the ordinates. The S_n weak form of the NTE is given with the following:

where:

• .

• .

Scattering Treatment

The scattering cross-section and phase function can be expanded in Legendre polynomials :

where . This results in the modified scattering kernel as shown below:

The spherical harmonics addition theorem can be applied to remove the Legendre polynomial :

where:

• and are the degree and maximum degree of the Legendre expansion.

• is the order of the spherical harmonics expansion.

• are the real spherical harmonics.

• are the moments of the group-to-group scattering phase function.

• are the moments of the group angular flux.

Substituting Eq. (2) into the scattering kernel yields the following:

In general the degree of the Legendre expansion corresponds to the degree of anisotropy in the medium.

Evaluating Flux Moments

The scattering expansion discussed above hinges on the evaluation of flux moments:

The integral over of the unit sphere (all directions) is equivalent to the spherical integral:

For functions symmetrical about it can be shown that this integral is equal to:

where the angular direction is a function of and . For functions which are not symmetrical about we can decompose the integral about such that both halves of the unit spheres are integrated completely:

This form allows us to evaluate angular integrals with a Gauss-Chebyshev product quadrature, where the polar integral is approximated with a Gauss-Legendre quadrature and the azimuthal integral is approximated with a Gauss-Chebyshev quadrature. This takes the form of:

where:

• is the number of Gauss-Legendre quadrature points.

• is the 'th Gauss-Legendre weight.

• is the 'th Gauss-Legendre point.

• is the number of Gauss-Chebyshev quadrature points.

• is the 'th Gauss-Chebyshev weight.

• is the 'th Gauss-Chebyshev point.

are the roots of the degree Legendre polynomial, and the weights are defined as:

and are:

The order of the quadrature is , as the angular flux is not guaranteed to be symmetrical about and must therefore be evaluated twice for positive and negative angular directions. We can chose the discrete ordinates such that:

Allowing for the use of the discrete ordinates solutions to evaluate the required flux moments.