discretize.TreeMesh.average_cell_to_face#

TreeMesh.average_cell_to_face#

Averaging operator from cell centers to faces (scalar quantities).

This property constructs an averaging operator that maps scalar quantities from cell centers to face. This averaging operator is used when a discrete scalar quantity defined cell centers must be projected to faces. Once constructed, the operator is stored permanently as a property of the mesh. See notes.

Returns:

(n_faces, n_cells) scipy.sparse.csr_matrix

The scalar averaging operator from cell centers to faces

Notes

Let ${\mathit{\varphi }}_{\mathit{c}}$ be a discrete scalar quantity that lives at cell centers. average_cell_to_face constructs a discrete linear operator ${\mathbf{A}}_{\mathbf{cf}}$ that projects ${\mathit{\varphi }}_{\mathit{c}}$ to faces, i.e.:

$${\mathit{\varphi }}_{\mathit{f}}={\mathbf{A}}_{\mathbf{cf}}{\mathit{\varphi }}_{\mathit{c}}$$

where ${\mathit{\varphi }}_{\mathit{f}}$ approximates the value of the scalar quantity at the faces. For each face, we are performing a weighted average between the values at adjacent cell centers. In 1D, where adjacent cells $i$ and $i+1$ have widths $h_i$ and $h_{i+1}$, $\varphi$ on face is approximated by:

$${\varphi }_{i+1/2}\approx \frac{h_{i+1}{\varphi }_i+h_i{\varphi }_{i+1}}{h_i+h_{i+1}}$$

On boundary faces, nearest neighbour is used to extrapolate the value from the nearest cell center. Once the operator is construct, the averaging is implemented as a matrix vector product, i.e.:

phi_f = Acf @ phi_c