Question

我使用以下MATLAB代码用隐式有限差分法实现沿杆的一维扩散方程。 `

xsize   =   10;                          % Model size, m
xnum    =   10;                          % Number of nodes
xstp    =   xsize/(xnum-1);              % Grid step
tnum    =   504;                         % number of timesteps
kappa   =   833.33;                      % Thermal diffusivity, m^2/s
dt     =   300;                          % Timestep
x       =   0:xstp:xsize;                %Creating vector for nodal point positions

tlbc = sin(linspace(0.1,2.9,tnum));      % left boundary condition

%Define initial temperature profile
tback   =   0;                           % background temperature, K

for i=1:1:xnum
    % Background profile  
    t0imp(i)    =   tback;               % profile for implicit solving
end
% Time cycle
timesum=0; % Elapsed time
for t=1:1:tnum

    % Matrix of coefficients initialization for implicit solving
    L       =       sparse(xnum,xnum);
    % Vector of right part initialization for implicit solving
    R       =       zeros(xnum,1);

    % Implicit solving of 1D temperature equation: dT/dt=kappa*d2T/dx2
    % Composing matrix of coefficients L()
    % and vector (column) of right parts R()
    % First point: T=tback
    L(1,1)  =       1;
    R(1,1)  =       tlbc(t);
    % Intermediate points
    for i=2:1:xnum-1
        % dT/dt=kappa*d2T/dx2
        % Tnew(i)/dt-kappa*(Tnew(i-1)-2*Tnew(i)+Tnew(i+1))/dx^2=Told(i)/dt

        L(i,i-1)    =   -kappa/xstp^2;
        L(i,i)      =   1/dt+2*kappa/xstp^2;
        L(i,i+1)    =   -kappa/xstp^2;
        R(i,1)      =   t0imp(i)/dt;
    end
    % Last point:T=tback
    L(xnum,xnum)    =   1;

    R(xnum,1)       =  tback;
    % Obtaining solution for implicit temperature profile
    t1imp           =   L\R;

end `

我想将右边界条件改为无限边界条件，以便10 m边界的存在对10 m域内的热流和温度流没有影响，好像在10 m处没有边界。

Answer 1

如果您有远离边界的局部现象，那么我建议您使用非均匀网格进行x空间离散化。网格应该在感兴趣的区域中具有更多的点。然后，您可以选择较大的xsize，但保持网格点的数量合理。

非均匀网格实现起来比较棘手（对于1-d来说不是那么糟糕）但幸运的是Matlab有pdepe来解决类型的非均匀方程

\开始{}方程 c \ left（x，t，u，\ frac {\ partial u} {\ partial x} \ right）\ frac {\ partial u} {\ partial t} = x ^ { - m} \ frac {\ partial} {\ partial t} \ left（x ^ mf \ left（x，t，u，\ frac {\ partial u} {\ partial x} \ right）\ right）+ S \ left（x，t，u，\ frac {\ partial u} {\ partial x} \ right） \ {端方程}

使用用户定义的网格物体。你可以在这里阅读更多关于pdepe的信息

https://www.mathworks.com/help/matlab/ref/pdepe.html

他们也解决了你感兴趣的等式。

用隐式有限差分法求解一维扩散方程的无穷远边界条件

1 个答案: