The Alternating Direction Implicit Method in 2-D

[phyzmatix]

Good day everybody!

In my Numerical Methods textbook (Applied Numerical Analysis, 7ed, Gerald and Wheatley) the authors derive two equations for the ADI method to be used in an iteration scheme. For row-wise traversions, they get

$$uO^{(k+1)}=uO^{(k)}+\rho(uL-2uO+uR)^{(k+1)}+\rho(uA-2uO+uB)^{(k)}$$

and for column-wise traversions

$$uO^{(k+2)}=uO^{(k+1)}+\rho(uA-2uO+uB)^{(k+2)}+\rho(uL-2uO+uR)^{(k+1)}$$

Where u represents the nodes, A, B, L, R are above, below, left and right respectively, O is the node in the centre (current), k represents the iteration and rho is an acceleration factor.

I understand that we alternate between these equations for successive iterations (hence the name ) but what I don't get is that it seems to me that the value we're trying to calculate is dependent on itself, e.g how do we determine $$uO^{(k+1)}$$ if we don't yet have $$(uL-2uO+uR)^{(k+1)}$$ ?

Any insight will be greatly appreciated!
phyz

 

[phyzmatix]

Anybody? If I didn't provide enough information, please let me know. If you have access to a worked example, please provide it and I'll figure it out for myself. Even if you have knowledge of a different method that uses a similar iteration scheme, that will also be handy.

 

[arildno]

Okay, I think I understand your predicament, namely that you have an in-part, implicit system, where quantities are, indeed, defined between themselves and other equally unknown quantities.

What you generally do then, for example for non-linear systems, is

1. that you have a method for determining a FIRST CHOICE for the unknown quantity that you put into the problematic right-hand side.

2. On basis of this, you calculate directly what the left-hand side should be (ideally, you are to get out the same quantity!).

3. You now go into a recursive loop:
Your newly calculated quantity is plugged into the right-hand side, calculating a new left-hand side quantity.

4. At the end of each step, you perform a quantity comparison analysis, that utilizes a necessary criterion for Cauchy convergence:
Namely:
That in order for some sequence to converge, the terms themselves must become equal to each other, i.e, their difference must go to zero as you proceed along your sequence.

5. You need some cut-off procedure, like for example:
If my 10 last terms do not vary more than 0.01% from each other, then I am SATISFIED, and say that the loop has converged.

Of course, this is NOT mathematically rigorous, but is how we do it in practice, and just about the only way to do so in practice.

Generally, a deep familiarity with the intricacies of stability and convergence is needed in order to set up really intelligent cut-off procedures.

 

[phyzmatix]

3. You now go into a recursive loop:
Your newly calculated quantity is plugged into the right-hand side, calculating a new left-hand side quantity.

Recursion! Of course! I sometimes forget that most (if not all) of these methods are designed specifically for use with computers!

Thanks arildno!

 

[blue_raver22]

guy's

Good day! The Alternating Direction Implicit (ADI) method is a popular numerical method used for solving partial differential equations (PDEs) in two-dimensional problems. It is often used in cases where other methods, such as the explicit method, may be unstable or inefficient.

The equations you have provided are indeed the ADI method's iteration scheme for row-wise and column-wise traversals. The method works by splitting the PDE into two simpler problems, one for the x-direction and one for the y-direction. This allows for easier and more stable calculations.

To address your concern, the value of uO^{(k+1)} is not solely dependent on itself, but it is also dependent on the values of uL, uR, uA, and uB. These values are calculated using the previous iteration's values of uO^{(k)} and the known boundary conditions. Therefore, the values of uO^{(k+1)} can be determined without having the values of (uL-2uO+uR)^{(k+1)} beforehand.

I hope this helps to clarify the concept of the ADI method for you. If you have any further questions, please don't hesitate to ask. Happy studying!