I use finite difference with periodic BCs on a square domain to discretize the Poisson equation. This problem is underdetermined and I need one more equation to solve, eg a dirichlet condition at one point. I can not find the corresponding square matrix.
It seems that the first derivatives have to satisfy the periodic boundary conditions as well. Maybe it'll help you.
Just consider that for Neumann BC.s you have to fulfill the compatibility relation to have one solution (apart the constant). If the relation is satisfied, you can also let that an iterative method set implicitly a constant value without the need to fix it a-priori. The value of the constant (that can be also a function of time) is not essential.
I usually use finite element for such kind of equation. from my understanding the periodic boundary are kind of redundant imposition of continuity conditions that already exist. I never impose them and obtain correct solution. I think you over constrained the problem, try without it.
Well, periodic condition implies that the surface integral of the dp/dn is zero, therefore the volume integral of the source term must be zero. This ensures the existance of a solution. This is a mathematical constraint, is not due to the type of discretization.
As I use ghost cells, I can impose Dirichlet on one cell at both ends, eg:
u(0,1) = - u(1,1)
u(n,1) = - u(n-1,1)
This way the solution is unique. Not sure about the periodic condition on this cell though.
Martin, no need to use ghost cells for periodic BC.s ...
again, if you satisfy the compatibility condition you get convergence in any iterative solver without fixing any value. Conversely, if you do not get convergence, it is likely you do not fulfill this requirement and fixing a value give you a convergence toward a possible erroneous solution.
Filippo, your solution works fine if I use an iterative solver.
What if I want to use a direct solver ? I need a non-singular matrix as an input ...
No problems if the rank of the matrix is correct, see
https://pdfs.semanticscholar.org/cebf/d7a3d72d7a7f3382b921abc8b04c39a36f95.pdf
https://www.researchgate.net/publication/279970840_SParC-LES_Enabling_large_eddy_simulations_with_parallel_sparse_matrix_computation_tools
Article SParC-LES: Enabling large eddy simulations with parallel spa...
Filippo, I have followed your idea. I used the PETSc library. Krylov solvers from PETSc can handle singular systems if we specify the null-space. It works fine !
Many thanks.
- Badges
- Science topic