Linear Source Approximation in MPACT for Efficient and Robust Multiphysics Whole-Core Simulations

Figures & data

Fig. 1. Depiction of the spatial and directional coordinate system used in the neutron transport equation.

Fig. 2. $E_3(\mathit{\tau })$ function calculation.

Algorithm 1 LIFA LSA algorithm for each MOC transport sweep

1: Prepare ${\bar{\varphi }}_{\mathit{li}}^{\mathit{rg},(\mathrm{\ell })}$, ${\widehat{\text{phii}}}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell })}$, and ${\psi }_{md}^{g,\mathrm{i}\mathrm{n},(\mathrm{\ell })}$ from previous sweep or initial guess

2: Calculate ${\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell })}$ in Eq. (12)(12) ${\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g}}=\frac{1}{4\mathit{\pi }}\left[\sum _{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}{\mathrm{\Sigma }}_{\mathit{s},\mathit{i}}^{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}\to \mathit{g}}{\widehat{\text{phii}}}_{\mathit{i}}^{\mathit{g}{ }^{\mathrm{\prime }}}+\frac{{\chi }_i^g}{k_{\mathrm{eff}}}\sum _{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}\mathit{\nu }{\mathrm{\Sigma }}_{\mathit{f},\mathit{i}}^{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}{\widehat{\text{phii}}}_{\mathit{i}}^{\mathit{g}{ }^{\mathrm{\prime }}}\right],$(12) for all $\mathit{i}$ and $\mathit{g}$

3: for Azimuthal angles $\mathit{a}$ do

4:     Calculate ${\bar{q}}_{\mathit{mi}}^{\mathit{g},(\mathrm{\ell })}$ in Eq. (10)(10) ${\bar{q}}_{\mathit{mi}}^{\mathit{g}}=\frac{1}{4\mathit{\pi }}\left[\sum _{\mathit{l}=0}^L\sum _{\mathit{r}=-\mathit{l}}^l{R}_{lm}^r\sum _{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}{\mathrm{\Sigma }}_{\mathit{s},\mathit{li}}^{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}\to \mathit{g}}{\bar{\varphi }}_{\mathit{li}}^{\mathit{rg}{ }^{\mathrm{\prime }}}+\frac{{\chi }_i^g}{k_{\mathrm{eff}}}\sum _{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}\mathit{\nu }{\mathrm{\Sigma }}_{\mathit{f},\mathit{i}}^{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}{\bar{\varphi }}_{\mathit{i}}^{\mathit{g}{ }^{\mathrm{\prime }}}\right],$(10) for all $\mathit{i}$, $\mathit{g}$, and corresponding $\mathit{p}$

5:     Calculate ${\hat{q}}_{\mathit{mi}}^{\mathit{g},(\mathrm{\ell })}$ in Eq. (20)(20) ${\hat{q}}_{\mathit{mi}}^{\mathit{g}}={\mathbf{\Omega }}_{\mathit{xy},\mathit{m}}\cdot {\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g}}/{\xi }_{\mathit{ai}}.$(20) for all $\mathit{i}$ and $\mathit{g}$

6:     for MOC rays do

7:         Calculate ${\bar{q}}_{\mathit{mki}}^{\mathit{g},(\mathrm{\ell })}$ in Eq. (19)(19) ${\bar{q}}_{\mathit{mki}}^{\mathit{g}}={\bar{q}}_{\mathit{mi}}^{\mathit{g}}+{\mathbf{x}}_{aki}^c\cdot {\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g}},$(19) for all $\mathit{k}$, $\mathit{g}$, and corresponding $\mathit{p}$

8:         for MOC segments $\mathit{k}$, polar angles $\mathit{p}$ and energy groups $\mathit{g}$ do

9:             Calculate $\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g},(\mathrm{\ell }+1)}$ in Eq. (28)(28) $\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}=\left({\psi }_{mki}^{g,in}-\frac{{\bar{q}}_{\mathit{mki}}^{\mathit{g}}}{{\mathrm{\Sigma }}_{t,i}^g}\right)E_1({\tau }_{mki}^g)-\frac{{\mathit{t}}_{\mathit{mki}}}{2}\frac{{\hat{q}}_{\mathit{mi}}^{\mathit{g}}}{{\mathrm{\Sigma }}_{t,i}^g}{E}_2({\tau }_{mki}^g),$(28)

10:             Update ${\psi }_{mki}^{g,\mathrm{o}\mathrm{u}\mathrm{t},(\mathrm{\ell }+1)}$ in Eq. (32)(32) ${\psi }_{mki}^{g,\mathrm{o}\mathrm{u}\mathrm{t}}={\psi }_{mki}^{g,\mathrm{i}\mathrm{n}}-{\tau }_{mki}^g\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}.$(32)

11:             Accumulate contribution to ${\bar{\varphi }}_{\mathit{li}}^{\mathit{rg},(\mathrm{\ell }+1)}$ in Eq. (34)(34) $\begin{array}{rl}& {\bar{\varphi }}_{\mathit{li}}^{\mathit{rg}}=\frac{1}{{\mathrm{\Sigma }}_{t,i}^g}\sum _m{\omega }_m{R}_{lm}^r{\bar{q}}_{\mathit{mi}}^{\mathit{g}}+\frac{1}{V_i{\mathrm{\Sigma }}_{t,i}^g}\sum _m{\omega }_m{R}_{lm}^r\\ & \sum _k\mathit{\delta }{A}_{a}sin{\theta }_p{\tau }_{mki}^g\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}.& (34)\end{array}$(34)

12:             Accumulate contribution to ${⟨\mathbf{x}{\psi }_{mki}^g(\mathit{t})⟩}_{\mathit{i}}^{(\mathrm{\ell }+1)}$ in Eq. (33)(33) $\begin{array}{rl}& {⟨\mathbf{x}{\psi }_{mki}^g(\mathit{t})⟩}_{\mathit{i}}=\frac{4\mathit{\pi }{\mathbf{M}}_i{\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g}}}{{\mathrm{\Sigma }}_{t,i}^g}+\frac{1}{{\mathrm{\Sigma }}_{t,i}^g{V}_i}\sum _m{\omega }_m\sum _k\mathit{\delta }{A}_{a}sin{\theta }_p\\ & [{\mathbf{x}}_{aki}^{\mathrm{i}\mathrm{n}}{\tau }_{mki}^g\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}+{\mathbf{\Omega }}_{\mathit{xy},\mathit{m}}\frac{{\mathit{t}}_{\mathit{mki}}}{{\xi }_{\mathit{ai}}}\\ & \left.\left(\frac{{\bar{q}}_{\mathit{mi}}^{\mathit{g}}}{{\mathrm{\Sigma }}_{t,i}^g}+\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}-{\psi }_{mki}^{g,\mathrm{o}\mathrm{u}\mathrm{t}}\right)\right].& (33)\end{array}$(33)

13:         end for

14:     end for

15:     Process contribution to ${\bar{\varphi }}_{\mathit{li}}^{\mathit{rg},(\mathrm{\ell }+1)}$

16: end for

17: Calculate ${\bar{\varphi }}_{\mathit{li}}^{\mathit{rg},(\mathrm{\ell }+1)}$ in Eq. (34)(34) $\begin{array}{rl}& {\bar{\varphi }}_{\mathit{li}}^{\mathit{rg}}=\frac{1}{{\mathrm{\Sigma }}_{t,i}^g}\sum _m{\omega }_m{R}_{lm}^r{\bar{q}}_{\mathit{mi}}^{\mathit{g}}+\frac{1}{V_i{\mathrm{\Sigma }}_{t,i}^g}\sum _m{\omega }_m{R}_{lm}^r\\ & \sum _k\mathit{\delta }{A}_{a}sin{\theta }_p{\tau }_{mki}^g\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}.& (34)\end{array}$(34)

18: Calculate ${⟨\mathbf{x}{\psi }_{mki}^g(\mathit{t})⟩}_{\mathit{i}}^{(\mathrm{\ell }+1)}$ in Eq. (33)(33) $\begin{array}{rl}& {⟨\mathbf{x}{\psi }_{mki}^g(\mathit{t})⟩}_{\mathit{i}}=\frac{4\mathit{\pi }{\mathbf{M}}_i{\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g}}}{{\mathrm{\Sigma }}_{t,i}^g}+\frac{1}{{\mathrm{\Sigma }}_{t,i}^g{V}_i}\sum _m{\omega }_m\sum _k\mathit{\delta }{A}_{a}sin{\theta }_p\\ & [{\mathbf{x}}_{aki}^{\mathrm{i}\mathrm{n}}{\tau }_{mki}^g\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}+{\mathbf{\Omega }}_{\mathit{xy},\mathit{m}}\frac{{\mathit{t}}_{\mathit{mki}}}{{\xi }_{\mathit{ai}}}\\ & \left.\left(\frac{{\bar{q}}_{\mathit{mi}}^{\mathit{g}}}{{\mathrm{\Sigma }}_{t,i}^g}+\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}-{\psi }_{mki}^{g,\mathrm{o}\mathrm{u}\mathrm{t}}\right)\right].& (33)\end{array}$(33)

19: Calculate ${\widehat{\text{phii}}}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell }+1)}$ in Eq. (14)(14) ${\mathbf{M}}_i{\widehat{\text{phii}}}_{\mathit{i}}^{\mathit{g}}={⟨\mathbf{x}{\psi }_{mki}^g(\mathit{t})⟩}_{\mathit{i}},$(14)

Algorithm 2 TCP₀ or P₀ LSA algorithm for each MOC transport sweep

1: Prepare ${\bar{\varphi }}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell })}$, ${\widehat{\text{phii}}}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell })}$, and ${\psi }_{md}^{g,\mathrm{i}\mathrm{n},(\mathrm{\ell })}$ from previous sweep or initial guess

2: Calculate ${\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell })}$ in Eq. (12)(12) ${\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g}}=\frac{1}{4\mathit{\pi }}\left[\sum _{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}{\mathrm{\Sigma }}_{\mathit{s},\mathit{i}}^{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}\to \mathit{g}}{\widehat{\text{phii}}}_{\mathit{i}}^{\mathit{g}{ }^{\mathrm{\prime }}}+\frac{{\chi }_i^g}{k_{\mathrm{eff}}}\sum _{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}\mathit{\nu }{\mathrm{\Sigma }}_{\mathit{f},\mathit{i}}^{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}{\widehat{\text{phii}}}_{\mathit{i}}^{\mathit{g}{ }^{\mathrm{\prime }}}\right],$(12) for all $\mathit{i}$ and $\mathit{g}$

3: Calculate ${\bar{q}}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell })}$ in Eq. (35)(35) ${\bar{q}}_{\mathit{i}}^{\mathit{g}}=\frac{1}{4\mathit{\pi }}\left[\sum _{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}{\mathrm{\Sigma }}_{\mathit{s},\mathit{i}}^{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}\to \mathit{g}}{\bar{\varphi }}_{\mathit{i}}^{\mathit{g}{ }^{\mathrm{\prime }}}+\frac{{\chi }_i^g}{k_{\mathit{eff}}}\sum _{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}\mathit{\nu }{\mathrm{\Sigma }}_{\mathit{f},\mathit{i}}^{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}{\bar{\varphi }}_{\mathit{i}}^{\mathit{g}{ }^{\mathrm{\prime }}}\right],$(35) for all $\mathit{i}$ and $\mathit{g}$

4: for Azimuthal angles $\mathit{a}$ do

5:     Calculate ${\hat{q}}_{\mathit{mi}}^{\mathit{g},(\mathrm{\ell })}$ in Eq. (20)(20) ${\hat{q}}_{\mathit{mi}}^{\mathit{g}}={\mathbf{\Omega }}_{\mathit{xy},\mathit{m}}\cdot {\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g}}/{\xi }_{\mathit{ai}}.$(20) for all $\mathit{i}$, $\mathit{g}$, and corresponding $\mathit{p}$

6:     for MOC rays do

7:         Calculate ${\bar{q}}_{\mathit{aki}}^{\mathit{g},(\mathrm{\ell })}$ in Eq. (36)(36) ${\bar{q}}_{\mathit{aki}}^{\mathit{g}}={\bar{q}}_{\mathit{i}}^{\mathit{g}}+{\mathbf{x}}_{aki}^{\mathrm{c}}\cdot {\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g}}.$(36) for all $\mathit{k}$ and $\mathit{g}$

8:         for MOC segments $\mathit{k}$, polar angles $\mathit{p}$ and energy groups $\mathit{g}$ do

9:             Calculate $\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g},(\mathrm{\ell }+1)}$ in Eq. (28)(28) $\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}=\left({\psi }_{mki}^{g,in}-\frac{{\bar{q}}_{\mathit{mki}}^{\mathit{g}}}{{\mathrm{\Sigma }}_{t,i}^g}\right)E_1({\tau }_{mki}^g)-\frac{{\mathit{t}}_{\mathit{mki}}}{2}\frac{{\hat{q}}_{\mathit{mi}}^{\mathit{g}}}{{\mathrm{\Sigma }}_{t,i}^g}{E}_2({\tau }_{mki}^g),$(28)

10:              Update ${\psi }_{mki}^{g,\mathrm{o}\mathrm{u}\mathrm{t},(\mathrm{\ell }+1)}$ in Eq. (32)(32) ${\psi }_{mki}^{g,\mathrm{o}\mathrm{u}\mathrm{t}}={\psi }_{mki}^{g,\mathrm{i}\mathrm{n}}-{\tau }_{mki}^g\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}.$(32)

11:              Accumulate contribution to ${\bar{\varphi }}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell }+1)}$ in Eq. (38)(38) $\begin{array}{rl}& {\bar{\varphi }}_{\mathit{i}}^{\mathit{g}}=\frac{1}{{\mathrm{\Sigma }}_{t,i}^g}\sum _m{\omega }_m{\bar{q}}_{\mathit{i}}^{\mathit{g}}+\frac{1}{V_i{\mathrm{\Sigma }}_{t,i}^g}\sum _m{\omega }_m\\ & \sum _k\mathit{\delta }{A}_{a}sin{\theta }_p{\tau }_{mki}^g\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}.& (38)\end{array}$(38)

12:              Accumulate contribution to ${⟨\mathbf{x}{\psi }_{mki}^g(\mathit{t})⟩}_{\mathit{i}}^{(\mathrm{\ell }+1)}$ in Eq. (37)(37) $\begin{array}{rl}& {⟨\mathbf{x}{\psi }_{mki}^g(\mathit{t})⟩}_{\mathit{i}}=\frac{4\mathit{\pi }{\mathbf{M}}_i{\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g}}}{{\mathrm{\Sigma }}_{t,i}^g}+\frac{1}{{\mathrm{\Sigma }}_{t,i}^g{V}_i}\sum _m{\omega }_m\\ & \sum _k\mathit{\delta }{A}_{a}sin{\theta }_p[{\mathbf{x}}_{aki}^{\mathrm{i}\mathrm{n}}\mathrm{\Delta }{\psi }_{mki}^g\\ & \left.+{\mathbf{\Omega }}_{\mathit{xy},\mathit{m}}\frac{{\mathit{t}}_{\mathit{mki}}}{{\xi }_{\mathit{ai}}}\left(\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}-{\psi }_{mki}^{g,\mathrm{o}\mathrm{u}\mathrm{t}}\right)\right].& (37)\end{array}$(37)

13:         end for

14:     end for

15: end for

16: Calculate ${\bar{\varphi }}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell }+1)}$ in Eq. (38)(38) $\begin{array}{rl}& {\bar{\varphi }}_{\mathit{i}}^{\mathit{g}}=\frac{1}{{\mathrm{\Sigma }}_{t,i}^g}\sum _m{\omega }_m{\bar{q}}_{\mathit{i}}^{\mathit{g}}+\frac{1}{V_i{\mathrm{\Sigma }}_{t,i}^g}\sum _m{\omega }_m\\ & \sum _k\mathit{\delta }{A}_{a}sin{\theta }_p{\tau }_{mki}^g\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}.& (38)\end{array}$(38)

17: Calculate ${⟨\mathbf{x}{\psi }_{mki}^g(\mathit{t})⟩}_{\mathit{i}}^{(\mathrm{\ell }+1)}$ in Eq. (37)(37) $\begin{array}{rl}& {⟨\mathbf{x}{\psi }_{mki}^g(\mathit{t})⟩}_{\mathit{i}}=\frac{4\mathit{\pi }{\mathbf{M}}_i{\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g}}}{{\mathrm{\Sigma }}_{t,i}^g}+\frac{1}{{\mathrm{\Sigma }}_{t,i}^g{V}_i}\sum _m{\omega }_m\\ & \sum _k\mathit{\delta }{A}_{a}sin{\theta }_p[{\mathbf{x}}_{aki}^{\mathrm{i}\mathrm{n}}\mathrm{\Delta }{\psi }_{mki}^g\\ & \left.+{\mathbf{\Omega }}_{\mathit{xy},\mathit{m}}\frac{{\mathit{t}}_{\mathit{mki}}}{{\xi }_{\mathit{ai}}}\left(\mathit{\psi }{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{mki}}^{\mathit{g}}-{\psi }_{mki}^{g,\mathrm{o}\mathrm{u}\mathrm{t}}\right)\right].& (37)\end{array}$(37)

18: Calculate ${\widehat{\text{phii}}}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell }+1)}$ in Eq. (14)(14) ${\mathbf{M}}_i{\widehat{\text{phii}}}_{\mathit{i}}^{\mathit{g}}={⟨\mathbf{x}{\psi }_{mki}^g(\mathit{t})⟩}_{\mathit{i}},$(14)

Fig. 3. Illustration of a locally negative LS.

Algorithm 3 LLSA algorithm for each MOC transport sweep

1: Prepare ${\bar{\varphi }}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell })}$ and ${\hat{\varphi }}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell })}$ from previous sweep or initial guess

2: for Region $\mathit{i}$ and energy groups $\mathit{g}$ do

3:     Calculate ${\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell })}$ in Eq. (12)(12) ${\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g}}=\frac{1}{4\mathit{\pi }}\left[\sum _{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}{\mathrm{\Sigma }}_{\mathit{s},\mathit{i}}^{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}\to \mathit{g}}{\widehat{\text{phii}}}_{\mathit{i}}^{\mathit{g}{ }^{\mathrm{\prime }}}+\frac{{\chi }_i^g}{k_{\mathrm{eff}}}\sum _{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}\mathit{\nu }{\mathrm{\Sigma }}_{\mathit{f},\mathit{i}}^{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}{\widehat{\text{phii}}}_{\mathit{i}}^{\mathit{g}{ }^{\mathrm{\prime }}}\right],$(12)

4:     Calculate ${\bar{q}}_{\mathit{i}}^{\mathit{g},min,(\mathrm{\ell })}$ in Eq. (40)(40) ${\bar{q}}_{\mathit{i}}^{\mathit{g},min}=\underset{\mathit{m},\mathit{k}}{min}\left\{{\bar{q}}_{\mathit{aki}}^{\mathit{g}}\pm {\hat{q}}_{\mathit{mi}}^{\mathit{g}}\frac{{\mathit{t}}_{\mathit{mki}}}{2}\right\}.$(40)

5:     Calculate ${\gamma }_i^{g,(\mathrm{\ell })}$ in Eq. (41)(41) ${\gamma }_i^g=min\left(\frac{{\bar{q}}_{\mathit{i}}^{\mathit{g}}}{{\bar{q}}_{\mathit{i}}^{\mathit{g}}-{\bar{q}}_{\mathit{i}}^{\mathit{g},min}},1\right).$(41)

6:     Multiply ${\gamma }_i^{g,(\mathrm{\ell })}$ to ${\hat{\mathbf{q}}}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell })}$

7:     if 2D/1D method is used then

8:         Calculate ${{\bar{q}}^{\prime }}_{\mathit{i}}^{\mathit{g},min,(\mathrm{\ell })}$ in Eq. (47)(47) ${{\bar{q}}^{\prime }}_{\mathit{i}}^{\mathit{g},min}=\left({\bar{q}}_{\mathit{i}}^{\mathit{g},min}-{\bar{q}}_{\mathit{i}}^{\mathit{g}}\right){\gamma }_i^g+{\bar{q}}_{\mathit{i}}^{\mathit{g}}.$(47)

9:         Obtain ${\bar{L}}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell })}$ in Eq. (44)(44) ${\bar{L}}_{\mathit{i}}^{\mathit{g}}=-\frac{{\bar{J}}_{\mathit{i},\mathit{z}+1/2}^{\mathit{g}}-{\bar{J}}_{\mathit{i},\mathit{z}-1/2}^{\mathit{g}}}{4\mathit{\pi }\mathrm{\Delta }{h}_z},$(44) from 1D axial nodal solver

10:         Perform TL splitting in Eq. (48)(48) $\begin{array}{rl}& \frac{d{\psi }_{mki}^g(\mathit{t})}{\mathit{dt}}+\left({\mathrm{\Sigma }}_{t,i}^g-\frac{{{\bar{q}}^{\prime }}_{\mathit{i}}^{\mathit{g},min}+{\bar{L}}_{\mathit{i}}^{\mathit{g}}}{{\bar{\varphi }}_{\mathit{i}}^{\mathit{g}}/4\mathit{\pi }}\right){\psi }_{mki}^g(\mathit{t})\\ & ={{\bar{q}}^{\prime }}_{\mathit{mki}}^{\mathit{g}}+{{\hat{q}}^{\prime }}_{\mathit{ai}}^{\mathit{g}}\left(\mathit{t}-\frac{{\mathit{t}}_{\mathit{mki}}}{2}\right)-{{\bar{q}}^{\prime }}_{\mathit{i}}^{\mathit{g},min}\\ & ={\bar{q}}_{\mathit{mi}}^{\mathit{g}}-{{\bar{q}}^{\prime }}_{\mathit{i}}^{\mathit{g},min}+{\gamma }_i^g{\mathbf{x}}_{aki}^{\mathrm{c}}\cdot {{\hat{\mathbf{q}}}^{\prime }}_{\mathit{i}}^{\mathit{g}}\\ & +{\gamma }_i^g\frac{{\mathbf{\Omega }}_{\mathit{xy},\mathit{m}}\cdot {{\hat{\mathbf{q}}}^{\prime }}_{\mathit{i}}^{\mathit{g}}}{{\xi }_{\mathit{ai}}}\left(\mathit{t}-\frac{{\mathit{t}}_{\mathit{mki}}}{2}\right).& (48)\end{array}$(48) if $({{\bar{q}}^{\prime }}_{\mathit{i}}^{\mathit{g},min,(\mathrm{\ell })}+{\bar{L}}_{\mathit{i}}^{\mathit{g},(\mathrm{\ell })})<0$

11:     end if

12: end for

13: Perform 2D MOC sweep for all $\mathit{z}$

Fig. 4. Geometry and materials of the 2D C5G7 benchmark.

TABLE I Mesh Options for the 2D C5G7 Benchmark

Parameters		Medium Mesh	Coarse Mesh
Fuel unit cell^a	Rod	3/8	1/4
	Coolant	4/16	1/8
Reflector unit cell^b		12$\times$12	3$\times$3

^aNumberof radial divisions/number of azimuthal divisions.

^bNumber of divisions in the $\mathit{x}$-direction $\times$ number of divisions in the $\mathit{y}$-direction.

TABLE II The Relative Number of Source Regions and MOC Segments of C5G7 Problem

Source Regions (Coarse/Medium)	MOC Segments (Coarse/Medium)
0.09	0.30

TABLE III Accuracy Comparisons for the C5G7 2D Benchmark Calculation Using Different Source Approximations and Meshes

Method	$\mathrm{\Delta }$$k_{\mathit{eff}}$ (pcm)^a	Pin Power Difference (%)
		RMS^b	Maximum
FS-Medium	2	0.19	0.76
LS-Medium	−7	0.13	0.50
FS-Coarse	65	0.81	2.19
LS-Coarse	−3	0.13	0.49

^aThe reference eigenvalue statistical error is $\pm$5 pcm.

^bThe reference RMS pin power statistical error is $\pm$0.14%.

Fig. 5. C5G7 problem pin power differences (in units of percent).

TABLE IV Performance Comparisons for the C5G7 2D Benchmark Calculation Using Different Source Approximations and Meshes

Method	Relative Iterations	Relative MOC Run Time/Iteration	Relative Memory
FS-Medium	1.00	1.00	1.00
LS-Medium	1.06	2.04	3.26
FS-Coarse	0.89	0.32	0.71
LS-Coarse	1.06	0.54	0.85

TABLE V Mesh Options for the 2D Fuel Assembly Problems

Parameters		Fine Mesh	Medium Mesh	Coarse Mesh
Fuel unit cell^a	UO2 pellet	15/32	3/8	2/1
	Gadolinia pellet	15/32	10/8	10/1
	Cladding	10/32	1/8	1/1
	Other annulus	1/32	1/8	1/1
	Coolant (PWR)	16/32	2/8	1/4
	Coolant (BWR)	16/32	6/32	2/8
BWR assembly gap unit cell^b		13$\times$10	5$\times$3	4$\times$3
Reflector unit cell^c		12$\times$12	4$\times$4	2$\times$2

^aNumber of radial divisions/number of azimuthal divisions.

^bNumber of divisions parallel to channel box/number of divisions perpendicular to channel box.

^cNumber of divisions in the $\mathit{x}$-direction $\times$ number of divisions in the $\mathit{y}$-direction.

Fig. 6. The relative number of source regions and MOC segments.

TABLE VI Case-Averaged Differences for Eigenvalue and Pin Power Distribution with TCP₀ Scattering

Method	Case-Averaged $\mathrm{\Delta }{k}_{\mathit{eff}}$ (pcm)	Case-Averaged Pin Power Difference (%)
		RMS	Maximum
FS-Medium	39	0.08	0.21
LS-Medium	23	0.04	0.13
FS-Coarse	163	0.33	0.71
LS-Coarse	24	0.05	0.16

TABLE VII Case-Averaged Difference for Eigenvalue and Pin Power Distribution with P₂ Scattering

Method	Case-Averaged $\mathrm{\Delta }{k}_{\mathit{eff}}$ (pcm)	Case-Averaged Pin Power Difference (%)
		RMS	Maximum
FS-Medium	40	0.12	0.27
LS-Medium	22	0.04	0.13
FS-Coarse	158	0.42	0.91
LS-Coarse	34	0.06	0.19

Fig. 7. Eigenvalue difference (compared to fine mesh case) for TCP₀ scattering.

Fig. 8. Pin power difference (compared to fine mesh case) for TCP₀ scattering.

Fig. 9. Eigenvalue difference compared to fine mesh calculation with P₂ scattering.

Fig. 10. Difference in pin power compared to fine mesh calculation with P₂ scattering.

TABLE VIII Case-Averaged Relative Run Time and Memory with TCP₀ Scattering

Method	Relative Iterations	Relative MOC Run Time/Iteration	Relative Memory
FS-Medium	1.00	1.00	1.00
LS-Medium	1.04	1.90	1.12
FS-Coarse	0.90	0.67	0.85
LS-Coarse	1.05	1.12	0.88

TABLE IX Case-Averaged Relative Run Time and Memory with P₂ Scattering

Method	Relative Iterations	Relative MOC Run Time/Iteration	Relative Memory
FS-Medium	1.00	1.00	1.00
LS-Medium	1.22	1.70	1.08
FS-Coarse	1.01	0.44	0.70
LS-Coarse	1.24	0.79	0.71

TABLE X VERA-5A 2D Core Problem Eigenvalue and Pin Power Results with TCP₀

Method	$\mathrm{\Delta }$$k_{\mathit{eff}}$ (pcm)	Pin Power Difference (%)
		RMS	Maximum
FS-Medium	−47	0.08	0.25
LS-Medium	−43	0.05	0.11
FS-Coarse	−57	0.58	1.39
LS-Coarse	−32	0.05	0.15

TABLE XI VERA-5A 2D Core Problem Eigenvalue and Pin Power Results with P₂

Method	$\mathrm{\Delta }$$k_{\mathit{eff}}$ (pcm)	Pin Power Difference (%)
		RMS	Maximum
FS-Medium	−44	0.12	0.34
LS-Medium	−29	0.07	0.18
FS-Coarse	−82	0.74	1.73
LS-Coarse	−23	0.07	0.21

Fig. 11. VERA-5A 2D core problem pin power differences with P₂ (in units of percent).

TABLE XII VERA-5A 2D Core Problem Relative Run Time and Memory with TCP₀

Method	Relative Iterations	Relative MOC Run Time/Iteration	Relative Memory
FS-Medium	1.00	1.00	1.00
LS-Medium	1.00	1.97	1.20
FS-Coarse	1.00	0.60	0.76
LS-Coarse	1.00	1.06	0.81

TABLE XIII VERA-5A 2D Core Problem Relative Run Time and Memory with P₂

Method	Relative Iterations	Relative MOC Run Time/Iteration	Relative Memory
FS-Medium	1.00	1.00	1.00
LS-Medium	1.07	1.68	1.12
FS-Coarse	1.00	0.41	0.56
LS-Coarse	1.00	0.69	0.59

TABLE XIV Burnup-Averaged Eigenvalue and Pin Power Differences for VERA 2A Depletion Problem

Method	Scattering	$\mathrm{\Delta }$$k_{\mathit{eff}}$ (pcm)	Pin Power Difference (%)
			RMS	Maximum
FS-Medium	TCP0	21	0.06	0.13
LS-Medium		40	0.05	0.12
FS-Coarse		29	0.10	0.20
LS-Coarse		27	0.06	0.17
FS-Medium	P2	11	0.06	0.14
LS-Medium		30	0.04	0.12
FS-Coarse		38	0.13	0.26
LS-Coarse		23	0.06	0.16

TABLE XV Burnup-Averaged Eigenvalue and Pin Power Differences for VERA 2P Depletion Problem

Method	Scattering	$\mathrm{\Delta }$$k_{\mathit{eff}}$ (pcm)	Pin Power Difference (%)
			RMS	Maximum
FS-Medium	TCP0	19	0.06	0.14
LS-Medium		33	0.04	0.12
FS-Coarse		89	0.16	0.39
LS-Coarse		23	0.05	0.15
FS-Medium	P2	17	0.07	0.15
LS-Medium		32	0.05	0.14
FS-Coarse		76	0.20	0.45
LS-Coarse		35	0.07	0.19

Fig. 12. VERA 2A depletion problem results.

Fig. 13. VERA 2P depletion problem results.

TABLE XVI VERA 2A Depletion Step-Averaged Relative Run Time and Memory with TCP₀ Scattering

Method	Relative Iterations	Relative MOC Run Time/Iteration	Relative Memory
FS-Medium	1.00	1.00	1.00
LS-Medium	1.00	1.84	1.07
FS-Coarse	1.00	0.66	0.88
LS-Coarse	1.00	1.07	0.90

TABLE XVII VERA 2P Depletion Step-Averaged Relative Run Time and Memory with TCP₀ Scattering

Method	Relative Iterations	Relative MOC Run Time/Iteration	Relative Memory
FS-Medium	1.00	1.00	1.00
LS-Medium	1.01	1.86	1.07
FS-Coarse	0.98	0.67	0.90
LS-Coarse	1.00	1.11	0.91

TABLE XVIII VERA 2A Depletion Step-Averaged Relative Run Time and Memory with P₂ Scattering

Method	Relative Iterations	Relative MOC Run Time/Iteration	Relative Memory
FS-Medium	1.00	1.00	1.00
LS-Medium	1.01	1.78	1.05
FS-Coarse	1.00	0.44	0.74
LS-Coarse	1.01	0.78	0.75

TABLE XIX VERA 2P Depletion Step-Averaged Relative Run Time and Memory with P₂ Scattering

Method	Relative Iterations	Relative MOC Run Time/Iteration	Relative Memory
FS-Medium	1.00	1.00	1.00
LS-Medium	1.03	1.73	1.04
FS-Coarse	0.97	0.42	0.75
LS-Coarse	1.02	0.78	0.76

TABLE XX VERA Problem 6 3D Single-Assembly Eigenvalue and Pin Power Results with TCP₀

Method	$\mathrm{\Delta }$$k_{\mathit{eff}}$ (pcm)	Pin Power Difference (%)		Fuel Temperature Difference (K)
		RMS	Maximum	RMS	Maximum
FS-Medium	$<$1	0.17	0.31	0.49	0.94
LS-Medium	−36	0.03	0.10	0.09	0.33
FS-Coarse	74	0.47	0.91	1.40	2.79
LS-Coarse	−20	0.06	0.17	0.22	0.61

TABLE XXI VERA Problem 6 3D Single-Assembly Eigenvalue and Pin Power Results with P₂

Method	$\mathrm{\Delta }$$k_{\mathit{eff}}$ (pcm)	Pin Power Difference (%)		Fuel Temperature Difference (K)
		RMS	Maximum	RMS	Maximum
FS-Medium	10	0.15	0.29	0.42	0.87
LS-Medium	−27	0.07	0.14	0.20	0.40
FS-Coarse	59	0.42	0.91	1.22	2.72
LS-Coarse	−34	0.07	0.19	0.23	0.68

TABLE XXII VERA Problem 6 Relative Run Time and Memory with TCP₀

Method	Relative Iterations	Relative MOC Run Time/Iteration	Relative Memory
FS-Medium	1.00	1.00	1.00
LS-Medium	0.91	1.81	1.09
FS-Coarse	1.00	0.71	0.86
LS-Coarse	0.91	1.07	0.88

TABLE XXIII VERA Problem 6 Relative Run Time and Memory with P₂

Method	Relative Iterations	Relative MOC Run Time/Iteration	Relative Memory
FS-Medium	1.00	1.00	1.00
LS-Medium	1.10	1.55	1.09
FS-Coarse	1.00	0.36	0.73
LS-Coarse	1.10	0.60	0.75

Fig. 14. Problem geometries.

TABLE XXIV Numerical Results from B&W 1484 Core I Simulations

Method	$\mathrm{\Delta }$$k_{\mathit{eff}}$ (pcm)	Maximum Difference in Pin Power (%)	Relative MOC Run Time	Relative Memory	Number of Cells with Negative LS^a
LSA	Reference	Reference	1.00	1.00	10
LLSA	$<$1	$<$0.01	1.05	1.00	0

^aThe total number of cells is 40 490.

TABLE XXV Numerical Results from PB2 Assembly Simulations

Method	$\mathrm{\Delta }$$k_{\mathit{eff}}$ (pcm)	Maximum Difference in Pin Power (%)	Relative MOC Run Time	Relative Memory	Number of Cells with Negative LS^a
LSA/split $\bar{q}+\bar{L}$	Reference	Reference	1.00	1.00	1260
LSA/split $\bar{L}$	$<$1	0.54	1.00	1.00	1260
LLSA	$<$1	0.06	1.06	1.00	0

^aThe total number of cells is 62 069.

Fig. 15. Comparisons between LSA and LLSA.