5.3.2 ADI Method for the Potential Equation

Alternating Direction Implicit technique: In addition to $\mbox{$\bf v$}$, construct another long vector $\mbox{$\bf w$}$ by linking together the columns of the matrix $\{ u_{i,j}\}$:

$$w_{s} = u_{i,j} \,,\;\;\;{\rm with}\;\;s=(j-1)N+i$$

Conversely,

$$j=\mbox{int} \left( \frac{s-1}{N}\right)+1 ; \;\; i=\left[ (s-1) \; \mbox{mod} \; N \right] + 1$$

The discretized potential equation is then

$$w_{s+1}-2w_{s}+w_{s-1}+v_{r+1}-2v_{r}+v_{r-1} = - (\Delta l)^{2} \rho_{i,j}$$

or

$$\mbox{${\bf A}$}_{1} \cdot \mbox{$\bf v$} + \mbox{${\bf A}$}_{2} \cdot \mbox{$\bf w$} = \mbox{$\bf b$}$$

$\mbox{${\bf A}$}_{1}$ acts on the ``rows'' of the $u_{i,j}$ lattice, $\mbox{${\bf A}$}_{2}$ effects the ``columns'' only:

Figure 5.6: ADI method

Both $\mbox{${\bf A}$}_{1}$ and $\mbox{${\bf A}$}_{2}$ are tridiagonal, and not pentadiagonal as the original matrix $\mbox{${\bf A}$}$. $\Longrightarrow$Recursion method



The optimal value of the relaxation parameter is given by

$$\omega = \sqrt{\lambda_{1} \lambda_{2}} \, ,$$

where $\lambda_{1}$ and $\lambda_{2}$ are the smallest and largest eigenvalue, respectively, of the matrix $\mbox{${\bf A}$}$. In the specific case of the potential equation, assuming a lattice with $M=N$, we have $\omega \approx \pi/N$.

EXERCISE: Apply the ADI method to the Laplace problem with $M=N=5$.