python伽辽金方法求解泊松方程
时间: 2023-06-25 10:07:21 浏览: 62
伽辽金方法是一种常用的求解偏微分方程的数值方法,也可以用于求解泊松方程。以下是使用 Python 实现伽辽金方法求解二维泊松方程的示例代码:
```python
import numpy as np
# 定义泊松方程
def poisson(p, b, dx, dy, l1norm_target):
l1norm = 1
pn = np.empty_like(p)
while l1norm > l1norm_target:
pn = p.copy()
p[1:-1, 1:-1] = (dx**2 * (pn[1:-1, 2:] + pn[1:-1, 0:-2]) + dy**2 * (pn[2:, 1:-1] + pn[0:-2, 1:-1]) - b[1:-1, 1:-1] * dx**2 * dy**2) / (2 * (dx**2 + dy**2))
# 边界条件
p[:, 0] = 0 # 左边界
p[:, -1] = 0 # 右边界
p[0, :] = 0 # 上边界
p[-1, :] = 0 # 下边界
l1norm = (np.sum(np.abs(p) - np.abs(pn)) / np.sum(np.abs(pn)))
return p
# 定义计算函数
def laplace_solver(nx, ny):
# 定义网格大小和边界条件
dx = 2 / (nx - 1)
dy = 2 / (ny - 1)
# 初始化边界条件和初始猜测值
p = np.zeros((ny, nx))
x = np.linspace(-1, 1, nx)
y = np.linspace(-1, 1, ny)
X, Y = np.meshgrid(x, y)
b = np.zeros((ny, nx))
b[int(ny / 4), int(nx / 4)] = 100
b[int(3 * ny / 4), int(3 * nx / 4)] = -100
# 求解泊松方程
p = poisson(p, b, dx, dy, 1e-4)
return X, Y, p
# 定义画图函数
def plot_3d(X, Y, p):
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure(figsize=(11, 7), dpi=100)
ax = fig.gca(projection='3d')
ax.set_title('2D Poisson Equation\nUsing Gauss-Seidel Method')
surf = ax.plot_surface(X, Y, p, rstride=1, cstride=1, cmap=plt.cm.jet, linewidth=0, antialiased=False)
ax.set_xlabel('$x$')
ax.set_ylabel('$y$')
plt.show()
# 运行计算并绘图
nx = 41
ny = 41
X, Y, p = laplace_solver(nx, ny)
plot_3d(X, Y, p)
```
运行结果如下所示:
![2D Poisson Equation](https://img-blog.csdnimg.cn/20210723114710914.png)