Tridiagonal system

The tridiagonal matrix to be solved is built using:

  

$$\text{odd } k \qquad A(k) \eta^{n+{\scriptscriptstyle \frac{1}{2}}}_{i\;j} + B(k) u^{n+1}_{i\;j} + C(k) \eta^{n+{\scriptscriptstyle \frac{1}{2}} }_{i\;j+1} = D(k)$$ (7.12)

$$\text{even } k \qquad A(k) u^{n+1}_{i\;j-1} + B(k) \eta^{n+{ {\scriptscriptstyle \frac{1}{2}}}}_{i\;j} + C(k) u^{n+1}_{i\;j} = D(k)$$ (7.13)

An iteration from k=k_i to k=k_f. 

  

$$\text{odd } k \begin{cases}A_k &= -\frac{g \Delta t}{2 \Delta x} \\ B_k &= 1 +... ...tstyle \frac{1}{2}} \Delta t \varepsilon \mathcal{D} u^n_{i\;j} \\ \end{cases}$$ (7.14)

$$\text{even } k \begin{cases}A_k &=-\frac{\Delta t}{2 \Delta x} H^n_{i\;j-1} \\ ... ...} v_{i\;j}^n - H^n_{i-{\scriptstyle \frac{1}{2}}\; j} v_{i-1\;j}^n) \end{cases}$$ (7.15)