有限差分法-热传导方程-显式法

2017-07-03 2017

整理了下有限差分法，为大变形固结有限差分的计算准备下。
 热传导方程如下：

初始条件:
 T=sin函数
 边界条件：
 左边界：0
 右边界：0
 长度1，
 时间1，
 传导系数1/4
 Matlab程序 

%%sd
%  Explicit Method 
clear; 
 % Parameters to define the heat equation and the range in space and time 
L = 1.;       %  Length of the wire 
T =1.;        %  Final time 
%  Parameters needed to solve the equation within the explicit method 
maxk = 2500;                 %  Number of time steps 
dt = T/maxk; 
n = 50;                      %  Number of space steps 
dx = L/n; 
cond = 1/4;                  %  Conductivity 
b = cond*dt/(dx*dx);     %  Stability parameter (b=<1) 
 %  Initial temperature of the wire: a sinus. 
for i = 1:n+1 
        x(i) =(i-1)*dx; 
        u(i,1) =sin(pi*x(i));         
end    
%  Temperature at the boundary (T=0) 
for k=1:maxk+1 
        u(1,k) = 0.; 
        u(n+1,k) = 0.; 
        time(k) = (k-1)*dt; 
end 
%  Implementation of the explicit method 
for k=1:maxk        %  Time Loop 
   for i=2:n;       %  Space Loop   
      u(i,k+1) =b*u(i+1,k)+(1-2*b)*u(i,k)+b*u(i-1,k);
      
     % u(i,k) + 0.5*b*(u(i-1,k)+u(i+1,k)-2.*u(i,k)); 
   end 
end 
% Graphical representation of the temperature at different selected times 
figure(1) 
plot(x,u(:,1),'-',x,u(:,100),'-',x,u(:,300),'-',x,u(:,600),'-') 
title('Temperature within the explicit method') 
xlabel('X') 
ylabel('T') 
 
figure(2) 
mesh(x,time,u') 
title('Temperature within the explicit method') 
xlabel('X') 
ylabel('Temperature')

结果：

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