利用Crank-Nicolson格式求解如下抛物线方程,给出详细推导过程(运用MATLAB编程) dμ/dt=μxx+μyy, 0≤x,y≤1. μ|δG=0 μ(x,y,0)=sin(πx)sin(πy).
时间: 2023-11-30 11:03:44 浏览: 83
首先,我们需要将抛物线方程离散化,使用Crank-Nicolson格式,得到:
$$\frac{\mu_{i,j}^{n+1} - \mu_{i,j}^n}{\Delta t} = \frac{1}{2}(\frac{\mu_{i+1,j}^{n+1} - 2\mu_{i,j}^{n+1} + \mu_{i-1,j}^{n+1}}{\Delta x^2} + \frac{\mu_{i,j+1}^{n+1} - 2\mu_{i,j}^{n+1} + \mu_{i,j-1}^{n+1}}{\Delta y^2} + \frac{\mu_{i+1,j}^{n} - 2\mu_{i,j}^{n} + \mu_{i-1,j}^{n}}{\Delta x^2} + \frac{\mu_{i,j+1}^{n} - 2\mu_{i,j}^{n} + \mu_{i,j-1}^{n}}{\Delta y^2})$$
其中,$\Delta t$,$\Delta x$,$\Delta y$ 分别表示时间步长,x方向和y方向的空间步长。
将上式整理,得到:
$$-\frac{\alpha}{2}\mu_{i-1,j}^{n+1} + (1+\alpha)\mu_{i,j}^{n+1} -\frac{\alpha}{2}\mu_{i+1,j}^{n+1} = \frac{\alpha}{2}\mu_{i-1,j}^{n} + (1-\alpha)\mu_{i,j}^{n} +\frac{\alpha}{2}\mu_{i+1,j}^{n} + \frac{\beta}{2}\mu_{i,j-1}^{n+1} + (1-\beta)\mu_{i,j}^{n+1} + \frac{\beta}{2}\mu_{i,j+1}^{n+1} + \frac{\beta}{2}\mu_{i,j-1}^{n} + (1-\beta)\mu_{i,j}^{n} + \frac{\beta}{2}\mu_{i,j+1}^{n}$$
其中,$\alpha = \frac{\Delta t}{\Delta x^2}$,$\beta = \frac{\Delta t}{\Delta y^2}$。
根据题目中的边界条件,我们可以得到:
$$\mu_{i,0}^{n} = \mu_{i,N}^{n} = \mu_{0,j}^{n} = \mu_{N,j}^{n} = 0$$
初始条件为:
$$\mu_{i,j}^{0} = \sin(\pi x_i)\sin(\pi y_j)$$
接下来,我们使用MATLAB编程来求解该方程。首先,我们需要定义一些参数和初始条件:
```matlab
% 定义时间步长,空间步长,以及网格数量
T = 0.5;
dt = 0.0001;
dx = 0.01;
dy = 0.01;
Nx = 101;
Ny = 101;
% 定义alpha和beta
alpha = dt / dx^2;
beta = dt / dy^2;
% 定义初始条件
mu = zeros(Nx, Ny);
for i = 1:Nx
for j = 1:Ny
mu(i,j) = sin(pi*(i-1)*dx)*sin(pi*(j-1)*dy);
end
end
```
接下来,我们可以使用循环来迭代求解:
```matlab
% 迭代求解
for t = 0:dt:T
% 边界条件
mu(1,:) = 0;
mu(Nx,:) = 0;
mu(:,1) = 0;
mu(:,Ny) = 0;
% 定义系数矩阵
A = zeros(Nx*Ny, Nx*Ny);
b = zeros(Nx*Ny, 1);
for i = 2:Nx-1
for j = 2:Ny-1
k = (i-1)*Ny + j;
A(k,k) = 1 + 2*alpha + 2*beta;
A(k,k-1) = -alpha;
A(k,k+1) = -alpha;
A(k,k-Ny) = -beta;
A(k,k+Ny) = -beta;
b(k) = mu(i,j)*((1-alpha)*(1-beta)+2*alpha+2*beta);
end
end
% 解线性方程组
mu_new = A \ b;
for i = 2:Nx-1
for j = 2:Ny-1
k = (i-1)*Ny + j;
mu(i,j) = mu_new(k);
end
end
end
```
最后,我们可以将结果可视化:
```matlab
% 可视化结果
[X,Y] = meshgrid(0:dx:1,0:dy:1);
surf(X,Y,mu');
```
完整代码如下:
```matlab
% 定义时间步长,空间步长,以及网格数量
T = 0.5;
dt = 0.0001;
dx = 0.01;
dy = 0.01;
Nx = 101;
Ny = 101;
% 定义alpha和beta
alpha = dt / dx^2;
beta = dt / dy^2;
% 定义初始条件
mu = zeros(Nx, Ny);
for i = 1:Nx
for j = 1:Ny
mu(i,j) = sin(pi*(i-1)*dx)*sin(pi*(j-1)*dy);
end
end
% 迭代求解
for t = 0:dt:T
% 边界条件
mu(1,:) = 0;
mu(Nx,:) = 0;
mu(:,1) = 0;
mu(:,Ny) = 0;
% 定义系数矩阵
A = zeros(Nx*Ny, Nx*Ny);
b = zeros(Nx*Ny, 1);
for i = 2:Nx-1
for j = 2:Ny-1
k = (i-1)*Ny + j;
A(k,k) = 1 + 2*alpha + 2*beta;
A(k,k-1) = -alpha;
A(k,k+1) = -alpha;
A(k,k-Ny) = -beta;
A(k,k+Ny) = -beta;
b(k) = mu(i,j)*((1-alpha)*(1-beta)+2*alpha+2*beta);
end
end
% 解线性方程组
mu_new = A \ b;
for i = 2:Nx-1
for j = 2:Ny-1
k = (i-1)*Ny + j;
mu(i,j) = mu_new(k);
end
end
end
% 可视化结果
[X,Y] = meshgrid(0:dx:1,0:dy:1);
surf(X,Y,mu');
```
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