Can someone help me with this Matlab code on 2D heat equation

I want to plot the temperature contour for the equation ∂T/∂t = ∂2T/∂x2 + ∂2T/∂y2,

The initial condition is taken to be t=0

It throws an error: Index in position 1 exceeds array bounds (must not exceed 1).

Here's my code:

>> nx=10;
>> ny=nx;
>> nt=100;
>> x=linspace(0,1,nx);
>> y=linspace(0,1,ny);
>> dx=x(2)-x(1);
>> dy=dx;
>> error=9e9;
>> tolerance=1e-4;
>> dt=1e-3;
>> T_L=0;
>> T_T=1;
>> T_R=0;
>> T_B=0;
>> T=300*ones(nx,ny);
>> T(2:ny-1,1)= T_L;
>> T(2:ny-1,nx)=T_R;
>> T(1,2:nx-1)=T_T;
>> T(ny,2:nx-1)=T_B;
>> T_old= T;
>> T_initial=0;
>> k1=1.1*(dt/(dx^2));
>> k2=1.1*(dt/(dy^2));
>> Jacobi_iteration=1;
>> for k= 1:nt
error=9e9;
while(error>tolerance)
for i=2:nx-1
for j= 2:ny-1
Term_1= 1/(1+(2*k1)+(2*k2));
Term_2= k1*Term_1;
Term_3= k2*Term_1;
H= (T_old(i-1,j))+(T_old(i+1,j));
V=(T_old(i,j+1))+(T_old(i,j-1));
T(i,j)= (T_initial(i,j)*Term_1)+ (H*Term_2)+(V*Term_3);
end
end
error=max(max(abs(T_old-T)));
T_old=T;
Jacobi_iteration= Jacobi_iteration+1;
end
T_intial=0;
%Plotting
figure(1)
contourf(x,y,T)
clabel(contourf(x,y,T))
colobar
colormap(jet)
set(gca,'ydir','reverse')
xlabel('X-Axis')
y-label('Y-Axis')
title(sprintf('No. of Unsteady Jacobi Iterations(Implicit)=%d', Jacobi_iteration));
pause(0.03)
end     

Answer 1

[Ravi Kumar]

Your T_initial is declared as a scalar, you are getting the error when you try to index into it beyond its size. Make that zeros of same size as T. You will find the corrected code below. On a related note, if you want to solve more general problems, check PDE Toolbox Heat Transfer workflow.

Regards,

Ravi

nx=10;

ny=nx;

nt=100;

x=linspace(0,1,nx);

y=linspace(0,1,ny);

dx=x(2)-x(1);

dy=dx;

error=9e9;

tolerance=1e-4;

dt=1e-3;

T_L=0;

T_T=1;

T_R=0;

T_B=0;

T=300*ones(nx,ny);

T(2:ny-1,1)= T_L;

T(2:ny-1,nx)=T_R;

T(1,2:nx-1)=T_T;

T(ny,2:nx-1)=T_B;

T_old= T;

T_initial=zeros(size(T));

k1=1.1*(dt/(dx^2));

k2=1.1*(dt/(dy^2));

Jacobi_iteration=1;

for k= 1:nt

error=9e9;

while(error>tolerance)

for i=2:nx-1

for j= 2:ny-1

Term_1= 1/(1+(2*k1)+(2*k2));

Term_2= k1*Term_1;

Term_3= k2*Term_1;

H= (T_old(i-1,j))+(T_old(i+1,j));

V=(T_old(i,j+1))+(T_old(i,j-1));

T(i,j)= (T_initial(i,j)*Term_1)+ (H*Term_2)+(V*Term_3);

end

end

error=max(max(abs(T_old-T)));

T_old=T;

Jacobi_iteration= Jacobi_iteration+1;

end

T_intial=0;

%Plotting

figure(1)

contourf(x,y,T)

clabel(contourf(x,y,T))

colorbar

colormap(jet)

set(gca,'ydir','reverse')

xlabel('X-Axis')

ylabel('Y-Axis')

title(sprintf('No. of Unsteady Jacobi Iterations(Implicit)=%d', Jacobi_iteration));

pause(0.03)

end

[Eshna Sasha]

A big thank you for the code. But after conducting 102 Jacobi iterations, I get this. Something is wrong for my case. Is it because of this? T=300*ones(nx,ny);