I am using Jacobi iterative method to solve sets of linear equations derived by discretization of governing equations of fluid. First I implemented First order Upwind method which works quite well but as soon as I implemented Second order Upwind, the program keep on running iteratively without solving a first time step. What could be the problem? What are the limitations of Jacobi method?
Hi Paresh
> What are the limitations of Jacobi method?
The Jacobi iterative method works fine with well-conditioned linear systems. If the linear system is ill-conditioned, it is most probably that the Jacobi method will fail to converge.
The Jacobi method can generally be used for solving linear systems in which the coefficient matrix is diagonally dominant.
> as soon as I implemented Second order Upwind, the program keep on running iteratively without solving a first time step.
Have you got a well-conditioned linear system when you moved to the second order case? Expected not.
It is clear that your termination condition was not met. That is the error norm (the norm of the difference between the solutions in two consequent iterations) could not be less than the given tolerance. If you monitor the error norm you shall notice that it is growing with the iterations.
If you are using Matlab, Octave or the Python numpy package, you can know the condition number of the coefficient matrix A by using the command: cond(A). Also, check the errors norms at the different iterations to understand the behavior of the solution with the Jacobi method.
Finally, I believe that you shall move to another numerical method for solving linear systems that can deal with the ill-conditioned systems. In Malab and Octave, the command:
x = A\b
will do everything on behalf of you.
I suggest you use Thomas Algorithm - Tri-Diagonal Matrix Algorithm.
http://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm
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