Homework 13

李尧 天眷班 2013301020048
problem 5.7
SOR_method Jacobi_method Laplace_equation

Abstrat
In this paper, two method to solving the Laplace equations will be introducted. Then there is some examples of numerical solutions of Laplace equation in different boundary condition. Finally, we will try to find the relation ship between iterating number and the number of grid elements.
Keywords:SOR method , Jacobi method , Laplace equation

Background
To solve realise physical problems, it's very important to solve partial differential equations. However, it's not always lucky to the exact solutions. But we have computers, and the numerical solutions definitely exist.

Jacobi method and SOR method
For the Laplace equation
$\nabla^2V=0$
can be written as the discrete form
$V(i,j,k)=\frac{V(i+1,j,k)+V(i-1，j，k)+V(i,j+1,k)+V(i,j-1,k)+V(i,j,k+1)+V(i,j,k-1)}{6}$
Similarily, the 2-D discrete form of Laplace equation is
$V(i,j,k)=\frac{V(i+1,j)+V(i-1,j)+V(i,j+1)+V(i,j-1)}{4}$
If the boundary condition is given, let computer caculate each element with the upper equation again and again. Suppose we finally get a stable result and this is the numerical solution.

In fact, there are some detail about how to iterate to be discussed.If we set two 2-D lists and the one store the recent result and the other store the old result. We always use the old to get the new one. This method is called Jacobi method.

Similarily to Jacobi method, but we just use one 2-D list. Mixcally use the new elements and the old elements. Let's define
$\Delta V=\frac{V_{old}(i+1,j)+V_{new}(i-1,j)+V_{old}(i,j+1)+V_{new}(i,j-1)}{4}-V_{old}(i,j)$
And get the new element by
$V_{new}(i,j)=V_{old}(i,j)+\alpha V$
$\alpha$ can be choosen in [1,2], and the best choice is
$\alpha=\frac{2}{1+\frac{\pi}{L}}$
L is the number of grid elements. this method is called SOR method

Program
program

Result
Remark:We take the output as the numerical solution if the sum of the absolute value of $\Delta V$ is lease than $10^{-13}$
Firstly, we'd show some example.
(1)A +1 square in field.

(2)Three +1 point in field

(3)A +1 line in field

(4)A wave condition

Then we compare the two method,

Conclusion
The iterating number of SOR method is around propotional to $L$, and
the iterating number of Jacobi method is around propotional to $L^2$