Column-wise j implicit sweep

For the j implicit sweep the approximation is:

 $$\begin{multline}\frac{\eta^{n+1}_{i\;j}-\eta^{n+{\scriptscriptstyle \frac{1}{2}}... ...j}) D_{oy} \eta^{n+{\scriptscriptstyle \frac{1}{2}}}_{i\;j} = 0 \end{multline}$$

 $$\begin{multline}\frac{v^{n+1}_{i\;j} - v^{n}_{i\;j}} {\Delta t} + {\scriptscript... ...tyle \frac{1}{2}}\;j}} - \varepsilon \mathcal{D} v^n_{i\;j} = 0 \end{multline}$$

$$u^{n+1}_{i+{\scriptscriptstyle \frac{1}{2}}\;j+{\scriptscriptstyl... ...c{1}{4} (u^{n+1}_{i\;j}+u^{n+1}_{i\;j+1}+u^{n+1}_{i-1\; j}+u^{n+1}_{i-1\; j+1})$$ (7.5)

$$H^{n+{\scriptscriptstyle \frac{1}{2}}}_{i+{\scriptscriptstyle \fr... ...tyle \frac{1}{2}}}_{i\;j} + \eta^{n+{\scriptscriptstyle \frac{1}{2}}}_{i+1\;j})$$ (7.6)

We use (4.6) at the center of the cell and (4.7) at the East and North of the cell ( $\uparrow{}$ in figure 4.1) to solve $\eta^{n+1}_{i\;j}$ and $v^{n+1}_{i\;j}$ for the j implicit sweep.