The 13th homework

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物基一班
陈天懿 2013301020146

Abstract

This article uses the Gauss-Seidel method to solve the Laplace equation, the compares the efficiency of it and the simultaneous over-relaxation method.

Background

In the space with no electric, the distribution of potential satisfies the well-known Laplace equation, that is:

$$\frac{{\partial}^2 V}{{\partial}x^2}+\frac{{\partial}^2 V}{{\partial}y^2}+\frac{{\partial}^2 V}{{\partial}z^2}=0$$
To solve equation, we use the approximation:
$$\begin{eqnarray}{} \frac{{\partial}^2 V}{{\partial}x^2}&=&\frac{1}{{\Delta}x}[\frac{V(i+1,j,k)-V(i,j,k)}{{\Delta}x}-\frac{V(i,j,k)-V(i-1,j,k)}{{\Delta}x}]\notag\\ &=&\frac{V(i+1,j,k)+V(i-1,j,k)-2V(i,j,k)}{{\Delta}x^2}\notag\\ \end{eqnarray}{}$$
Applay this procedure to all three partial derivatives we get:
$$V(i,jk)=\frac{1}{6}[V(i+1,j,k)+V(i-1,j,k)+V(i,j+1,k)+V(i,j-1,k)+V(i,j,k+1)+V(i,j,k-1)]$$

Main

The potential disstribution of the point electric


Boundry conditions for the point electric is none, means it has the free boundry condition.

The potential disstribution of the parallel-plate capacitor

parallel-plate capacitor is: +1 on the left plate, -1 on the right plate

Comparison of the efficiency of the Gauss-Seidel method and the SOR method
Theoritically, the iteration steps needed for the Gauss-Seidel method is proportional to the square of the length of lattice. The SOR method is only proportional to the length. We can show this directly by the picture below:

So the SOR method can increase the efficiency of calculation rapidly.

Conclusion

The validity of the numerical simulation of the distribution of electric potential is conformed for the results correspond with our knowledge of electromagnetism. And the SOR method is much more efficient than the Gauss-Seidel method.

Reference

Computational Physics-Nicholas J. Giordano, Hisao Nakanishi