4.2. The Method of Nearby Problems for k-Eigenvalue Problems

The method of nearby problems can be seamlessly applied to fixed source neutron transport problems. This is due to the presence of the external source term in (1) and the simple addition of the discrete residual. In \(k\)-eigenvalue problems, there is no external source term present given the nature of the criticality problems [LM93]. The discrete residual must be included in the criticality problem to solve for the nearby flux utilizing a technique different from adding it to the fixed source problems. The solver used for calculating the neutron flux for \(k\)-eigenvalue problems must be adjusted to allow for the incorporation of the residual source term.

The \(k\)-eigenvalue neutron transport equation is introduced in the neutron transport chapter with (4). Using the power iteration, the numerical angular flux and numerical \(\keff\) value are calculated. The curve fit flux \(S_{m, g}\) is calculated in the same manner as in the fixed-source formulation. For the analytical curve fit, the associated curve fit \(\keff^{*}\) term can be calculated as

\[\keff^{*} = \left[ \sum_{i=1}^{I} \left(\chi_{g} \sum_{g'=1}^{G} \nu_{g'} \sig[g' \rightarrow g]{f} \overline{S}_{g'} \right) \right] \left[ \sum_{i=1}^{I} \left(\bsOmega \cdot \nabla S_{m, g} + \sig[g]{t} S_{m, g} - \sum_{g'=1}^{G} \sig[g' \rightarrow g]{s} \overline{S}_{g'} \right) \right]^{-1},\]

which can be used for the \(\keff\) error calculation. The fission source term \(q_{g}^{*}\),

\[q_{g}^{*} = \frac{1}{\keff^{*}} \chi_{g} \sum_{g'=1}^{G} \nu_{g'} \sig[g' \rightarrow g]{f} \overline{S}_{g'},\]

is used to replace the fission and external source terms in (2) to calculate the continuous residual. The modified fission source term does not have an angular dependence, meaning the related spatially continuous \(R_{g}\) and discrete \(r_{g}\) residuals do not have angular dependence.

Algorithm — k-Eigenvalue Method of Nearby Problems Power Iteration of the Multigroup Discrete Ordinates Equation

\[\begin{split}\begin{array}{l} \textbf{Require: material properties } (\sig[g]{t}, \sig[g'\to g]{s}, \chi_g, \nu_{g'}, \sig[g'\to g]{f}), \\ \quad \text{initial guess } (\widetilde{\phi}), \text{ discretization parameters } (\bsOmega_m, w_m), \\ \quad \text{curve fit parameters } (\overline{S}_g, \keff^{*}), \text{ residual } (r_g), \text{ tolerance } \varepsilon_K \\[2pt] \Delta_K \gets 1 + \varepsilon_K, \quad k \gets 1 \\ \phi{}^{(0)} \gets \widetilde{\phi} \\ \keff{}^{(1/2)} \gets \keff^{*} \left[ \displaystyle\sum_{i=1}^{I} \left(\chi_g \displaystyle\sum_{g'=1}^{G} \nu_{g'} \sig[g'\to g]{f} \phi_{g'}^{(0)} \right) \right] \left[ \displaystyle\sum_{i=1}^{I} \left(\chi_g \displaystyle\sum_{g'=1}^{G} \nu_{g'} \sig[g'\to g]{f} \overline{S}_{g'} \right) \right]^{-1} \\ \textbf{while } \Delta_K > \varepsilon_K \textbf{ do} \\ \quad q = \dfrac{1}{\keff{}^{(k-1)}} \chi_g \displaystyle\sum_{g'=1}^{G} \nu_{g'} \sig[g'\to g]{f} \phi_{g'}{}^{(k-1)} + r_g \\ \quad \phi{}^{(k)} \gets \text{source-iteration algorithm with } q_g \gets q,\ \widetilde{\phi} \gets \phi{}^{(k-1)} \qquad \triangleright\ \text{Source iteration} \\ \quad \keff{}^{(k)} = \dfrac{\keff{}^{(k-1)} \chi_g \sum_{g'=1}^{G} \nu_{g'} \sig[g'\to g]{f} \phi_{g'}{}^{(k)}}{\chi_g \sum_{g'=1}^{G} \nu_{g'} \sig[g'\to g]{f} \phi_{g'}{}^{(k-1)}} \qquad \triangleright\ \text{Update } \keff \\ \quad \phi{}^{(k)} \gets \phi{}^{(k)} \left[\displaystyle\sum_{i=1}^{I} \sum_{g=1}^{G} \left(\phi{}^{(k)}\right)^2\right]^{-1/2} \qquad \triangleright\ \text{Normalize flux} \\ \quad \Delta_K \gets \left\| \dfrac{\phi{}^{(k)} - \phi{}^{(k-1)}}{\phi{}^{(k)}} \right\|_2 \\ \quad k \gets k + 1 \\ \textbf{end while} \\ \textbf{return } \phi \gets \phi_g{}^{(k)}, \quad \keff \gets \keff{}^{(k)} \end{array}\end{split}\]

To incorporate the discretized residual \(r_{g}\) into the criticality calculation in (4), the power iteration method must be altered. The standard power iteration, explained in the power-iteration algorithm, is changed to allow for the residual term to be added to each iteration. This can be seen in the MNP power-iteration algorithm, which utilizes the method developed by [WMC18] including the residual at every outer iteration. Another important difference is that the \(\keff\) value is initialized with the curve fit scalar flux \(\overline{S}_{g}\) and \(\keff^{*}\) values. This is indicated by \(\keff{}^{(1/2)}\) to show that it comes after the angular flux initialization and is consistent with the nomenclature of [WMC18]. Employing the modified power iteration with the residual and curve fit parameters, the nearby angular flux \(\psi\mnp\) and nearby \(k\)-effective value \(\keff\mnp\) can be calculated. The spatial discretization error between the nearby flux and the curve fit flux can be used with (5) to ensure that the numerical solution is sufficiently discretized.

The modified power iteration needed to run the method of nearby problems leads to a more involved approach when using this technique with the Monte Carlo MC/DC solver [MVP+24]. This is not a topic that has been investigated in this analysis, and the Monte Carlo solver is used with the fixed source problems. The \(k\)-eigenvalue problem for Monte Carlo based solvers is an area for future work. When applying the method of nearby problems to criticality problems, only a discrete ordinates solver is employed.