[K.C.]

Dear Foamers,

I am trying to use the fvMatrix-class for solving a sparse equation system in my model that can not be written in terms of fvm::laplacian oder fvm::div. Therfore I tried to understand the implementation of fvm::laplacian and fvm::div, to get confident with upper(), lower() and diag().

During my study I completly understood fvm::laplacian, but i get in trouble with fvm::div.
Imagine a simple 1D mesh with a simple advection of concentration by velocity U (phi on faces). Using Finite-Volume-Method for cell number 2 (so directly between cell 1 and cell 3), I would come up with the equations and discretisation in line 1 to 3 of the attached figure (weights for surface-interpolation are 0.25 and 0.75 to make the difference clear).


But using the negSumDiag() in fvm::div

for (register label face=0; face<l.size(); face++)
{
    Diag[l[face]] -= Lower[face];
    Diag[u[face]] -= Upper[face];
}

would end up with the matrix in line 4 of the attached figure.


I don't understand, how this can be the correct implementation of Finite-Volume-Method for the divergence-operator. Obviously it is correct, but I can not see my error in writing down the equations and the matrix. Thanks for every reply.

[youmengtear]

Originally Posted by K.C.

Dear Foamers,

I am trying to use the fvMatrix-class for solving a sparse equation system in my model that can not be written in terms of fvm::laplacian oder fvm::div. Therfore I tried to understand the implementation of fvm::laplacian and fvm::div, to get confident with upper(), lower() and diag().

During my study I completly understood fvm::laplacian, but i get in trouble with fvm::div.
Imagine a simple 1D mesh with a simple advection of concentration by velocity U (phi on faces). Using Finite-Volume-Method for cell number 2 (so directly between cell 1 and cell 3), I would come up with the equations and discretisation in line 1 to 3 of the attached figure (weights for surface-interpolation are 0.25 and 0.75 to make the difference clear).


But using the negSumDiag() in fvm::div

for (register label face=0; face<l.size(); face++)
{
    Diag[l[face]] -= Lower[face];
    Diag[u[face]] -= Upper[face];
}

would end up with the matrix in line 4 of the attached figure.


I don't understand, how this can be the correct implementation of Finite-Volume-Method for the divergence-operator. Obviously it is correct, but I can not see my error in writing down the equations and the matrix. Thanks for every reply.

Maybe, in the second equation the first div term show be negative, which should conside the direction of the face between 1 and 2.

[K.C.]

Thanks for your answer, youmengtear.

Do you mean the sign of phi_i_j?
phi_i_j is defined as u_face*n_face, where n_face is the vector pointing outward the cell (not mesh.Sf()). So phi_i_j could be negative but that would not solve the problem.

When "phi_i_j*(c_i*weight + c_j*(1-weight))" is formed into a matrix equation, phi_i_j could not transfrom the "+" of interpolation into a "-", while the other C_value is not transformed, I guess.

[youmengtear]

Originally Posted by K.C.

Thanks for your answer, youmengtear.

Do you mean the sign of phi_i_j?
phi_i_j is defined as u_face*n_face, where n_face is the vector pointing outward the cell (not mesh.Sf()). So phi_i_j could be negative but that would not solve the problem.

When "phi_i_j*(c_i*weight + c_j*(1-weight))" is formed into a matrix equation, phi_i_j could not transfrom the "+" of interpolation into a "-", while the other C_value is not transformed, I guess.

In openfoam, Sf has the direction from owner to neighbour. And in internal the owner has lower label, while neighbour has upper one. So not the sign of phi_i_j but the sign of Sf means.