exercise 10

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背景

1.Laplace'equation
In regions of space that do not contain any electric charges,the electric potential obeys Laplace's equation

$$\frac{\partial^2V}{\partial{x}^2}+\frac{\partial^2V}{\partial{y}^2}+\frac{\partial^2V}{\partial{z}^2}=0$$ After some calculation,we find $$V(i,j,k)=\frac 16[V(i+1,j,k)+V(i-1.j.k)+V(i,j+1,k)$$ $$+V(i,j-1,k)+V(i,j,k+1)+V(i,j,k-1)]$$ In terms of $$E_x=-\frac{\partial{V}}{\partial{x}}$$ We can use a symmetric form for the derivative $$E_x(i,j)\approx-\frac{V(i+1,j)-V(i-1,j)}{2\Delta{x}}$$
For Jacobi method,the equation can be expressed as $$V_{new}(i,j)=\frac 14[V_{old}(i+1,j)+V_{old}(i-1,j)$$ $$+V_{old}(i,j+1)+V_{old}(i,j-1)]$$ For SOR method, $$V(i,j)=\frac 14[V(i-1,j)+V(i+1,j)+V(i,j-1)+V(i,j+1)]$$ Let $V^{\star}(i,j)$ be the new value of the potential .Then,we think of $$\Delta{V(i,j)}=V^{\star}(i,j)-V_{old}(i,j)$$ $$V_{new}(i,j)=\alpha\Delta{V(i,j)}+V_{old}(i,j)$$ The best choice for $\alpha$ is $$\alpha\approx\frac{2}{1+\pi/L}$$
2.Poisson'equation
$$V(i,j,k)=\frac 16[V(i-1,j,k)+V(i+1,j,k)+V(i,j+1,k)+$$ $$V(i,j-1,k)+V(i,j,k+1)+V(i,j,k-1)]+\frac{\rho(i,j,k)\Delta{x^2}}{6}$$

正文

1.中间为一矩形等势面



2.中间有四个极板，电势为+1，-0.5，+0.5，-1



3.正方体中央放一点电荷



4.数值计算与Jacobi方法的比较

距离越远，与理论计算偏差越大

代码

code1
code2

致谢

参考13级郭潇的作业