21 where we have defined b$_{j}$$_{+}$ 1 2 = ( b$_{j}$ + b$_{j}$$_{+1}$ ) / 2 and $^{b$_{j}$}$− 1 2 = ( $^{b$_{j}$}$− 1 + b$_{j}$ ) / 2. This can be easily generalize to three di- mensions as in which Then the discrete version of Eq. (A3) is Then the Newton-Gauss-Seidel iteration says that we can obtain a new (and often more accurate) solution of u , u new $_{i,j,k}$, using our knowledge about the old (and less accu- rate) solution u old i,j,k as The old solution will be replaced by the new solution to u$_{i,j,k}$ once the new solution is ready, using the red-black Gauss-Seidel sweeping scheme. Note that u new i,j,k = u old i,j,k − L h ( u old i,j,k ) ∂L h ( u old i,j,k ) /∂u$_{i,j,k}$ . (A11) ∂L $^{h}$( u$_{i,j,k}$ ) ∂u$_{i,j,k}$ = 1 2 h 2 e u$_{i,j,k}$ (1 + e $^{u$_{i,j,k}$}$) 2 [ u$_{i}$$_{+1}$$_{,j,k}$ + $^{u$_{i}$}$− 1 ,j,k + u$_{i,j}$$_{+1}$$_{,k}$ + $^{u$_{i,j}$}$− 1 ,k + u$_{i,j,k}$$_{+1}$ + $^{u$_{i,j,k}$}$− 1 − 6 u$_{i,j,k}$ ] − 1 2 h 2 [ b$_{i}$$_{+1}$$_{,j,k}$ + $^{b$_{i}$}$− 1 ,j,k + b$_{i,j}$$_{+1}$$_{,k}$ + $^{b$_{i,j}$}$− 1 ,k + b$_{i,j,k}$$_{+1}$ + $^{b$_{i,j,k}$}$− 1 + 6 b$_{i,j,k}$ ] − ( H$_{0}$B ) 2 ac 2 3 γ $^{2}$Ω$_{CDM}$ ρ CDM $_{c,i,j,k}$(1 + e $^{u$_{i,j,k}$}$) γ b$_{i,j,k}$ − ( H$_{0}$B ) 2 ac 2 3 µβ $^{2}$Ω$_{V}$$_{0}$ a $^{3}$(1 + e $^{u$_{i,j,k}$}$) β [ 1 − (1 + e $^{u$_{i,j,k}$}$) β ] µ +1 b$_{i,j,k}$ [ 1 + ( µ + 1) (1 + e $^{u$_{i,j,k}$}$) β 1 − (1 + e $^{u$_{i,j,k}$}$) β ] . (A12) L $^{h}$( u$_{i,j,k}$ ) = 1 h 2 [ b$_{i}$$_{+}$ 1 $_{$_{2}$,j,k}$u$_{i}$$_{+1}$$_{,j,k}$ − u$_{i,j,k}$ ( b$_{i}$$_{+}$ 1 $_{2}$,j,k + $^{b$_{i}$}$− 1 $_{2}$,j,k ) + $^{b$_{i}$}$− 1 $^{$_{$_{2}$,j,k}$u$_{i}$}$− 1 ,j,k ] + 1 h 2 [ b$_{i,j}$$_{+}$ 1 $_{$_{2}$,k}$u$_{i,j}$$_{+1}$$_{,k}$ − u$_{i,j,k}$ ( b$_{i,j}$$_{+}$ 1 $_{2}$,k + $^{b$_{i,j}$}$− 1 $_{2}$,k ) + $^{b$_{i,j}$}$− 1 $^{$_{$_{2}$,k}$u$_{i,j}$}$− 1 ,k ] + 1 h 2 b$_{i,j,k}$$_{+}$ 1 $_{2}$u$_{i,j,k}$$_{+1}$ − u$_{i,j,k}$ b$_{i,j,k}$$_{+}$ 1 2 + $^{b$_{i,j,k}$}$− 1 2 + $^{b$_{i,j,k}$}$− 1 $^{$_{2}$u$_{i,j,k}$}$− 1 − ( H$_{0}$B ) 2 ac 2    3 γ Ω$_{CDM}$ ρ CDM $_{c,i,j,k}$(1 + e $^{u$_{i,j,k}$}$) γ + 3 µβ Ω$_{V}$$_{0}$ a $^{3}$(1 + e $^{u$_{i,j,k}$}$) β [ 1 − (1 + e $^{u$_{i,j,k}$}$) β ] µ +1    + ( H$_{0}$B ) 2 ac 2 [ 3 γ Ω$_{CDM}$ e $_{γ}$√$_{κ}$$_{¯}$ ϕ + 3 µβ Ω$_{V}$$_{0}$ a $^{3}$e $_{β}$√$_{κ}$$_{¯}$ ϕ [ 1 − e $_{β}$√$_{κ}$$_{¯}$ ϕ ] µ +1 ] . (A10) L $^{h}$( u$_{i,j,k}$ ) = 0 , (A9) ∇ · ( b ∇ u ) → 1 h 2 [ b$_{i}$$_{+}$ 1 $_{$_{2}$,j,k}$u$_{i}$$_{+1}$$_{,j,k}$ − u$_{i,j,k}$ ( b$_{i}$$_{+}$ 1 $_{2}$,j,k + $^{b$_{i}$}$− 1 $_{2}$,j,k ) + $^{b$_{i}$}$− 1 $^{$_{$_{2}$,j,k}$u$_{i}$}$− 1 ,j,k ] + 1 h 2 [ b$_{i,j}$$_{+}$ 1 $_{$_{2}$,k}$u$_{i,j}$$_{+1}$$_{,k}$ − u$_{i,j,k}$ ( b$_{i,j}$$_{+}$ 1 $_{2}$,k + $^{b$_{i,j}$}$− 1 $_{2}$,k ) + $^{b$_{i,j}$}$− 1 $^{$_{$_{2}$,k}$u$_{i,j}$}$− 1 ,k ] + 1 h 2 b$_{i,j,k}$$_{+}$ 1 $_{2}$u$_{i,j,k}$$_{+1}$ − u$_{i,j,k}$ b$_{i,j,k}$$_{+}$ 1 2 + $^{b$_{i,j,k}$}$− 1 2 + $^{b$_{i,j,k}$}$− 1 $^{$_{2}$u$_{i,j,k}$}$− 1 . (A8)