CHAPTER 10: Elliptic PDEs Lieberman Method: Part 1 of 2   

In this segment we will talk about how to solve second order linear elliptic partial differential equations by using the Lieberman method. We are showing it in two parts. The first part will be concerned with the theory and the second part will be concerned with the example. If you want more details about this particular method all you have to do is go to the numerical methods web page, click on keyword, and click on elliptic partial differential equations. So again, what we are doing is we will be using the same physical example of using an elliptic partial differential equation where we are seeing that hey, we want to be able to find out what the temperature in this plate is. It's a rectangular plate where it can be a plate which it doesn't have to be necessarily rectangular. So just to keep the algebra simple we are assuming a rectangular plate. The temperatures at the four edges are given to us and we want to find out what the steady state temperature will be inside this particular plate. And the steady state temperature in this plate is governed by this Laplacian equation, a second order linear elliptic partial differential equation and we need to figure out how can we find out the temperatures at different points in this particular plate as we are trying to solve this particular problem numerically. So we have talked about that we had discretized the equation here, the Laplacian equation by writing down this as our discretized equation of the Laplace by simply saying that hey, we have this node here at we can write down the temperatures of that node in terms of the temperatures of the surrounding nodes which are this node, this one, this one, and this one. And of course, we are also assuming the delta x is the same as delta y. That's how we get an equation which looks simplified as this. If delta x is not equal to delta y then we can still write down the equation in discretized form, but then delta x and delta y will also be part of the equation. So if you recall again the discretixed equation is given as this. The Gauss-Seidel method asks you to rewrite the equation for the temperature at node i,j in terms of the temperatures at the other 4 nodes which I just showed to you. And for the Gauss-Seidel method, this particular equation is solved iteratively for each and every node at whcih we don't know what the temperature, steady state temperature is and we keep on iterating it until we find that the pre-specified tolerance is met. So what is the difference in the Lieberman method? The difference in the Lieberman method is that we still use the Gauss-Seidel method but what we are doing is that we are going to use, we are going to give ratings to the new temperature which we just found out and the temperature which we have previously, from the previous situation. So, we will take a value of lambda and multiple it to the temperature which we just found out and we will add to it (1-lambda) times the temperature which we found out earlier, in the previous iteration. So the value of lambda, which is generally chosen is between 1 and 2 and that's why it's called over-relaxation because you are going to a lot more weight to the temperature which you have found now. You are going to multiply the temperature which you have found now by a number which is between 1 and 2 and the previous temperature is going to get a weightage of 1 minus lambda. It is going to be a negative number there and the reason why we are doing that is for one simple reason. We want to accelerate the convergence of the, of finding these temperatures and the reason why we are able to take such a, able to take such a route is because the coefficient matrix for this particular problem turns out to be diagonally dominant. And since it is diagonally dominant we know that the Gauss-Seidel methiod is guaranteed to convergence and we accelerate this convergence process by choosing a lambda to be between 1 and 2. If we choose a lambda to be less than 1, that's also considered in some cases where you may find out that the, the process might be a little bit divergent and in order to make that particular process convergent, you under-relax by choosing a lambda which is a number which is less than 1. So that's what the Leiberman method is all about that it is the same as the Gauss-Seidel method except for the fact that we revise our temperature estimate by giving a weightage to the current temperature, the temperature from the current iteration and the temperature from the previous iteration. And that's the end of this segment and we will take an example in the next part of this segment.