Multigroup discretisationΒΆ

If we take the standard neutron transport equation:

\[\begin{split}\Omega \cdot \nabla \psi(\mathbf{r}, \Omega, E) + &\sigma(\mathbf{r}, \Omega) \psi(\mathbf{r}, \Omega, E) \\ &=\iint \sigma_s(\mathbf{r}, \Omega' \to \Omega, E' \to E) \psi(\mathbf{r}, \Omega', E') dE' d\Omega' + Q(\mathbf{r}, \Omega, E)\end{split}\]

and integrate it over energy using the following assumptions:

  • Cross sections are assumed to be constant and independent of the solution.

  • Angular dependence of total cross section is ignored (arising from integrating over a angularly dependent angular flux).

  • The infinite sum over scattering order \(\ell\) is truncated to a finite maximum value \(L\). Even for highly anisotropic scattering (important in fusion due to the high energy neutrons), relatively low values are acceptable: the standard FENDL fusion nuclear data library provides \(L=6\).

it can be shown (see here for a detailed derivation) that the resulting equation is:

\[\begin{split}\left[\Omega \cdot \nabla + \sigma_g(\mathbf{r})\right]\psi_g(\mathbf{r}, \Omega) = \sum_{\ell}\sum_{\tilde{g}}Y_{\ell}(\Omega) \sigma^s_{\ell g \tilde{g}}(\mathbf{r})\int_{S^2}Y_{\ell} (\tilde{\Omega})&\psi_{\tilde{g}}(\mathbf{r},\tilde{\Omega}) \,d \tilde{\Omega} \nonumber \\ &+ q_g(\mathbf{r}, \Omega) \ \ \ \forall g\end{split}\]

where \(\psi_{g}(\mathbf{r}, \Omega) = \int_{E_g}^{E_{g-1}} \psi(\mathbf{r}, \Omega, E) dE\) is the group-integrated angular flux, \(Q_g(\mathbf{r}, \Omega) = \int_{E_g}^{E_{g-1}} Q(\mathbf{r}, \Omega, E) dE\) is the group-integrated external source, \(\sigma_{g}(\mathbf{r}, \Omega)\) is the group-averaged total cross section, \(Y_{\ell}(\Omega)\) are the real spherical harmonics (\(\ell = (l,m)\)), and \(\sigma^s_{\ell, g, g' }(\mathbf{r})\) are the Legendre moments of the group-averaged scattering cross section from \(g'\) to \(g\).

By convention, we take the energy groups such that \(E_0\) is the highest energy and \(E_{G}\) is the lowest energy.

Thus, the multigroup transport equation is a discrete set of coupled spatial-angular PDEs, one for each energy group.