solving a heat diffusion problem, using Crank-Nicolson AND gauss-Seidel method

vagelis Karvounis · June 5, 2012, 14:36

Hello everyone!

First of all i 'm not sure if i am posting on the right forum! May someone advise me about that!

Here is what i am dealing with. A heat diffusion problem on an aluminum plate. The plate is represented by a grid of points. Crank-Nicolson method gives me an equation to calculate each point's temperature by using the temperatures of the surrounding points.

T(i,j)=A*((a*dt/2)*(T'(i-1,j)-2*T'(i,j)+T'(i+1,j)+T(i-1,j)+T(i+1,j))/dx^2 +(a*dt/2)*(T'(i,j-1)-2*T'(i,j)+T'(i,j+1)+T(i,j-1)+T(i,j+1))/dy^2+ T'(i,j))

where T' is the temperature of a point at the previous time point, dt is the time between two time points and a is a heat diffusion constant of aluminum. dx,dy is the spatial division for the grid in x and y axis respectively

also A=1/(1+dt*a*(1/dx^2)+dt*a*(1/dy^2))

T'(i,j) must be given, as also some boundary conditions must be given too.
But the equation uses T(i-1,j),T(i+1,j),T(i,j-1),T(i,j+1) which are unknown.
At this point Gauss-Seidel method is useful. I can use an arbitrary value for these unknown T values and by iterating the calculations for each point, T(i,j) are converging to the right values! Of course after that, T values are placed on T' and the process is repeated to calculate temperatures for the next time point.

I am writing a code on matlab for this problem. The code is very slow.Trying to deal with making it faster, i found, that the shorter is the time step(dt) the less are the iterations i need to succeed a convergence on Gauss-Seidel. For my code, i use a 101X301 points grid and if dt=0.01sec it takes about 15 iterations to succeed convergence, more than 50 iterations if dt=0.1sec and more than 500 if dt=1sec.

Could someone guide me to find out how the convergence process is being affected by dt or anything else?

Thank you very much!

FMDenaro · June 5, 2012, 15:00

when you reduce the time step is absolutely normal and sound phjysical that the number of iterations decreases ... you simply have a physical state at the new time step that is closer to the old one....
Try using a SOR method, is faster than GS.

Prudhvi · March 25, 2015, 02:30

Hey vagelis,

I am trying to solve 2D heat equation using Crank Nicolson implementing gauss siedel method. my grid size is 128*128. time step dt=0.1. code is very slow in matlab. I am not able to get results quickly. please let me know if i can do anything to increase my execution time.

beer · March 25, 2015, 09:44

You could try using a different linear solver. Matlab has an implementation of the bi-conjugated gradient method (bicgstab) included.
http://de.mathworks.com/help/matlab/ref/bicgstab.html
The link explaines how to use it. The conjugate gradient methods are heaps faster than the gauss-seidel, jacobi etc solvers.

June 5, 2012, 14:36	solving a heat diffusion problem, using Crank-Nicolson AND gauss-Seidel method	#1
vagelis Karvounis New Member Vagelis Karvounis Join Date: May 2012 Posts: 1 Rep Power: 0	Hello everyone! First of all i 'm not sure if i am posting on the right forum! May someone advise me about that! Here is what i am dealing with. A heat diffusion problem on an aluminum plate. The plate is represented by a grid of points. Crank-Nicolson method gives me an equation to calculate each point's temperature by using the temperatures of the surrounding points. T(i,j)=A((adt/2)(T'(i-1,j)-2T'(i,j)+T'(i+1,j)+T(i-1,j)+T(i+1,j))/dx^2 +(adt/2)(T'(i,j-1)-2T'(i,j)+T'(i,j+1)+T(i,j-1)+T(i,j+1))/dy^2+ T'(i,j)) where T' is the temperature of a point at the previous time point, dt is the time between two time points and a is a heat diffusion constant of aluminum. dx,dy is the spatial division for the grid in x and y axis respectively also A=1/(1+dta(1/dx^2)+dta*(1/dy^2)) T'(i,j) must be given, as also some boundary conditions must be given too. But the equation uses T(i-1,j),T(i+1,j),T(i,j-1),T(i,j+1) which are unknown. At this point Gauss-Seidel method is useful. I can use an arbitrary value for these unknown T values and by iterating the calculations for each point, T(i,j) are converging to the right values! Of course after that, T values are placed on T' and the process is repeated to calculate temperatures for the next time point. I am writing a code on matlab for this problem. The code is very slow.Trying to deal with making it faster, i found, that the shorter is the time step(dt) the less are the iterations i need to succeed a convergence on Gauss-Seidel. For my code, i use a 101X301 points grid and if dt=0.01sec it takes about 15 iterations to succeed convergence, more than 50 iterations if dt=0.1sec and more than 500 if dt=1sec. Could someone guide me to find out how the convergence process is being affected by dt or anything else? Thank you very much!

June 5, 2012, 15:00		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,882 Rep Power: 73	when you reduce the time step is absolutely normal and sound phjysical that the number of iterations decreases ... you simply have a physical state at the new time step that is closer to the old one.... Try using a SOR method, is faster than GS.

March 25, 2015, 02:30		#3
Prudhvi New Member prudhvi Join Date: Oct 2014 Posts: 1 Rep Power: 0	Hey vagelis, I am trying to solve 2D heat equation using Crank Nicolson implementing gauss siedel method. my grid size is 128*128. time step dt=0.1. code is very slow in matlab. I am not able to get results quickly. please let me know if i can do anything to increase my execution time.

March 25, 2015, 09:44		#4
beer Member Join Date: Dec 2012 Posts: 92 Rep Power: 13	You could try using a different linear solver. Matlab has an implementation of the bi-conjugated gradient method (bicgstab) included. http://de.mathworks.com/help/matlab/ref/bicgstab.html The link explaines how to use it. The conjugate gradient methods are heaps faster than the gauss-seidel, jacobi etc solvers.