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Steady state heat equation:-((∂^2 T)/(∂x^2 )+(∂^2 T)/(∂y^2 )=-Q(x,y)

  1. Aug 5, 2014 #1
    Hi friends i am trying to solve heat equation for a single time domain,with Finite Difference Method. I wrote a program in MATLAB,it sucessfully worked, but as i increse the number of points, my program is taking lots of time to give out result.

    Can anybody help me in increasing the speed of my Program:-


    clear;
    close all;
    clc;
    n = 5;% n grids and has n - 2 interior points per dimension

    x = linspace(0,1,n);% line in x direction of 0 to 1 and divided into n parts

    dx = x(2)-x(1); % distance between grids

    y = x;

    dy = dx;

    k=1; %thermal conductivity

    Q=300 ; % heat source
    m=(n-2)^2;

    %Right hand side matrix

    B=zeros(m,1);

    if (mod(m,2)==0)
    B(m/2,1)=Q;
    else
    B((m+1)/2,1)=Q;
    end

    %Main Matrix

    Tsol = zeros(n);
    %boundry conditions
    Tsol(1,1:n) = 0; %TOP
    Tsol(n,1:n) = 0; %BOTTOM
    Tsol(1:n,1) = 0; %LEFT
    Tsol(1:n,n) = 0; %RIGHT

    tic
    error = 1; z = 0;
    while error > 0.000001
    z = z+1;
    c=1;
    Torg = Tsol;
    for i = 2:n-1
    for j = 2:n-1
    Tsol(i,j) =abs(((dx^2*B(c,1))+((Torg(i+1,j)+Torg(i-1,j)+Torg(i,j+1)+Torg(i,j-1))*k))/(4*k));
    c=c+1;
    end
    end

    error = max(max(abs(Torg-Tsol)));
    end


    toc

    %% plotting

    figure, contour(x,y,Tsol),
    title('Temperature (Steady State)'),xlabel('x'),ylabel('y'),colorbar



    figure,pcolor(x,y,Tsol),shading interp,
    title('Temperature (Steady State)'),xlabel('x'),ylabel('y'),colorbar
     
  2. jcsd
  3. Aug 13, 2014 #2
    I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
     
  4. Aug 20, 2014 #3
    Are you sure it even works at it stands now? What I think you might be trying to do is to solve the heat equation with a source term in the middle of a 2d system and with zero dirichlet boundary condition. Is that correct? Because that is not what I see when I change n.
     
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