[natsukashi_heiwa]

Hello everybody

Currently, I am engrossed in a university assignment that involves conducting a benchmark with various solvers such as Jacobi, Gauss-Seidel, and the Conjugate Gradient for solving the Poisson pressure equation.

Now, both the Jacobi and Gauss-Seidel Solvers can be employed in component form. This means you don't solve them as a matrix-vector product but rather with classical two nested loops, iterating through each component. If the domain is discretized as nxn, then the complexity of the component-form corresponds to O(n^2). In this form, the pressure field is expressed as a matrix.

On the flip side, the matrix-form necessitates a matrix-vector product (Au=f). In this setup, the pressure field is expressed as a vector, and the matrix A represents the discretized Laplacian operator, where I use FDM for simplicity reasons.

The challenge I'm currently grappling with is converting an already coded Lid Driven Cavity from component-form to matrix-form. The issue arises because when converting a pressure matrix-field (nxn) to a matrix-form, the matrix A becomes substantial (nxn, nxn) since the pressure vector field becomes of size nxn. Even though the matrix-vector product of Au=f is O(n^2), the sheer size of n in the matrix-form leads to a significant slowdown in my simulation.

So, for Navier-Stokes on FDM, particularly in the case of the Lid Driven Cavity, should one avoid the matrix-form for the solvers? If so, how can the Conjugate Gradient compete with the component form? The challenge lies in the fact that the Conjugate Gradient cannot be used in component-form since its nature resides in vectors (Krylov spaces).

In conclusion, I suspect there might be a critical aspect I've overlooked, and any guidance or insights would be greatly appreciated. 🙈

[LuckyTran]

The matrix is sparse


Indeed if the matrix was full there would never be any need to apply any linearization and to just brute force it.

[natsukashi_heiwa]

Originally Posted by LuckyTran

The matrix is sparse


Indeed if the matrix was full there would never be any need to apply any linearization and to just brute force it.

I see... right!!!!!

Having a sparse matrix, one does not need to do the full matrix-vector product but can instead do a sparse matrix-vector product and the size of the compressed sparse matrix is not (nxn,nxn) but (depth,nxn) where depth is the number of items in a row, which in 2D corresponds do 3 because the discretised laplacian has three elements (-1, 2, -1)

Thank you very much for your help, it really brought clarity to my mind.

Best regards

[FMDenaro]

The variables we solve are always the components of a vector, thus formally the vector for the pressure is (Nx1) and you can link the topology to the computational structured grid in a simple way in which the n-th component is associated to the i,j,k node.
A table describing such association is in the textbook of Ferziger, Peric and Street. This way, you can work solving the formal product matrix-vector in a simple instruction of the components p(n) multiplied by scalar values.