采用adi格式求解一维热方程

采用ADI（Alternating Direction Implicit）格式求解一维热方程可以提高计算稳定性和计算速度，以下是Python代码示例：

import numpy as np
import matplotlib.pyplot as plt

# 设置参数
L = 1 # 杆的长度
T = 1 # 总时间
alpha = 1 # 热扩散系数
N = 100 # 空间网格数
M = 1000 # 时间网格数
dx = L/N # 空间步长
dt = T/M # 时间步长
r = alpha*dt/dx**2 # 热传导数

# 初始化温度分布
u = np.zeros((N+1,M+1))
u[:,0] = np.sin(np.pi*np.linspace(0,L,N+1)) # 初始温度分布

# 设置矩阵
A1 = np.eye(N-1,N-1)*(1+2*r)
A1 += np.eye(N-1,N-1,k=1)*(-r)
A1 += np.eye(N-1,N-1,k=-1)*(-r)
B1 = np.eye(N-1,N-1)*(1-2*r)
B1 += np.eye(N-1,N-1,k=1)*r
B1 += np.eye(N-1,N-1,k=-1)*r
A2 = np.eye(N-1,N-1)*(1+2*r)
A2 += np.eye(N-1,N-1,k=1)*(-r)
A2 += np.eye(N-1,N-1,k=-1)*(-r)
B2 = np.eye(N-1,N-1)*(1-2*r)
B2 += np.eye(N-1,N-1,k=1)*r
B2 += np.eye(N-1,N-1,k=-1)*r

# 进行时间迭代
for k in range(M):
    # 第一步
    u[1:N,k+1/2] = np.linalg.solve(A1,np.dot(B1,u[1:N,k]))
    u[0,k+1/2] = u[1,k+1/2]
    u[N,k+1/2] = u[N-1,k+1/2]
    # 第二步
    u[1:N,k+1] = np.linalg.solve(A2,np.dot(B2,u[1:N,k+1/2]))
    u[0,k+1] = u[1,k+1]
    u[N,k+1] = u[N-1,k+1]

# 绘制温度随时间变化的图像
x = np.linspace(0,L,N+1)
t = np.linspace(0,T,M+1)
X,T = np.meshgrid(x,t)
fig = plt.figure()
ax = fig.gca(projection='3d')
surf = ax.plot_surface(X, T, u.T, cmap='coolwarm', linewidth=0, antialiased=False)
ax.set_xlabel('Length')
ax.set_ylabel('Time')
ax.set_zlabel('Temperature')
fig.colorbar(surf, shrink=0.5, aspect=5)
plt.show()

该代码使用了ADI格式来求解一维热传导方程，并绘制了温度随时间变化的三维图像。