2D heat equation using gauss-seidel method

[strythe]

Hi guys. Badly need some help. We were given this 2D heat equation BV problem. On the square plate, values on all four edges are given (2 are Neumann, 2 are Dirichlet). And we are to solve this problem using FDM, on a 5point stencil.

So I used FDM approximations to derive the formula for the temperature at the nodes in terms of its adjacent nodes. What I got was a matrix formulation: [Aij][Tj]=[Cj] where A is the coefficient matrix, T --> variables, C --> constants. So I was able to solve for the temperature at the nodes. Then, we have to iterate the whole Gauss Seidel process until a certain %relative error is achieved.

How do I proceed with the iteration after the first one? Since from what I can think of, the first temperature values obtained doesn't change the matrix formulation. Or am I wrong?

 

[jvicens]

Hello there,

Thank you for reaching out for help with your 2D heat equation BV problem. It sounds like you have made good progress so far by deriving the formula for the temperature at the nodes using FDM approximations and setting up a matrix formulation.

To answer your question about the iteration process, you are correct that the first temperature values obtained will not change the matrix formulation. However, the temperature values will change after each iteration, which will then affect the matrix formulation in the next iteration. This means that you will need to update the matrix formulation and solve for the temperature values at each iteration until the desired %relative error is achieved.

To proceed with the iteration process, you can follow these steps:

1. Start with an initial guess for the temperature values at the nodes. This can be any reasonable value, such as the average of the boundary temperatures.

2. Use the matrix formulation you derived to solve for the temperature values at the nodes.

3. Update the matrix formulation using the new temperature values obtained from the previous step.

4. Repeat steps 2 and 3 until the desired %relative error is achieved.

It is important to note that the number of iterations required to achieve the desired %relative error will depend on factors such as the size of the problem, the boundary conditions, and the accuracy of your initial guess. You may need to experiment with different initial guesses and/or increase the number of iterations to achieve the desired accuracy.

I hope this helps you proceed with your iteration process. If you have any further questions or need clarification, please don't hesitate to ask. Good luck with your problem!