property CylindricalMesh.face_x_divergence#

X-derivative operator (x-faces to cell-centres).

This property constructs a 2nd order x-derivative operator which maps from x-faces to cell centers. The operator is a sparse matrix \(\mathbf{D_x}\) that can be applied as a matrix-vector product to a discrete scalar quantity \(\boldsymbol{\phi}\) that lives on x-faces; i.e.:

dphi_dx = Dx @ phi

For a discrete vector whose x-component lives on x-faces, this operator can also be used to compute the contribution of the x-component toward the divergence.

(n_cells, n_faces_x) scipy.sparse.csr_matrix

The numerical x-derivative operator from x-faces to cell centers


Below, we demonstrate 1) how to apply the face-x divergence operator, and 2) the mapping of the face-x divergence operator and its sparsity. Our example is carried out on a 2D mesh but it can be done equivalently for a 3D mesh.

We start by importing the necessary packages and modules.

>>> from discretize import TensorMesh
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> import matplotlib as mpl

For a discrete scalar quantity \(\boldsymbol{\phi}\) defined on the x-faces, we take the x-derivative by constructing the face-x divergence operator and multiplying as a matrix-vector product.

>>> h = np.ones(40)
>>> mesh = TensorMesh([h, h], "CC")

Create a discrete quantity on x-faces

>>> faces_x = mesh.faces_x
>>> phi = np.exp(-(faces_x[:, 0] ** 2) / 8** 2)

Construct the x-divergence operator and apply to vector

>>> Dx = mesh.face_x_divergence
>>> dphi_dx = Dx @ phi

Plot the original function and the x-divergence

>>> fig = plt.figure(figsize=(13, 6))
>>> ax1 = fig.add_subplot(121)
>>> w = np.r_[phi, np.ones(mesh.nFy)]  # Need vector on all faces for image plot
>>> mesh.plot_image(w, ax=ax1, v_type="Fx")
>>> ax1.set_title("Scalar on x-faces", fontsize=14)
>>> ax2 = fig.add_subplot(122)
>>> mesh.plot_image(dphi_dx, ax=ax2)
>>> ax2.set_yticks([])
>>> ax2.set_ylabel("")
>>> ax2.set_title("X-derivative at cell center", fontsize=14)
>>> plt.show()

(Source code, png, pdf)


The discrete x-face divergence operator is a sparse matrix that maps from x-faces to cell centers. To demonstrate this, we construct a small 2D mesh. We then show the ordering of the elements in the original discrete quantity \(\boldsymbol{\phi}}\) and its x-derivative \(\partial \boldsymbol{\phi}}/ \partial x\) as well as a spy plot.

>>> mesh = TensorMesh([[(1, 6)], [(1, 3)]])
>>> fig = plt.figure(figsize=(10, 10))
>>> ax1 = fig.add_subplot(211)
>>> mesh.plot_grid(ax=ax1)
>>> ax1.plot(
...     mesh.faces_x[:, 0], mesh.faces_x[:, 1], "g>", markersize=8
... )
>>> for ii, loc in zip(range(mesh.nFx), mesh.faces_x):
...     ax1.text(loc[0]+0.05, loc[1]+0.02, "{0:d}".format(ii), color="g")
>>> ax1.plot(
...     mesh.cell_centers[:, 0], mesh.cell_centers[:, 1], "ro", markersize=8
... )
>>> for ii, loc in zip(range(mesh.nC), mesh.cell_centers):
...     ax1.text(loc[0]+0.05, loc[1]+0.02, "{0:d}".format(ii), color="r")
>>> ax1.set_xticks([])
>>> ax1.set_yticks([])
>>> ax1.spines['bottom'].set_color('white')
>>> ax1.spines['top'].set_color('white')
>>> ax1.spines['left'].set_color('white')
>>> ax1.spines['right'].set_color('white')
>>> ax1.set_xlabel('X', fontsize=16, labelpad=-5)
>>> ax1.set_ylabel('Y', fontsize=16, labelpad=-15)
>>> ax1.set_title("Mapping of Face-X Divergence", fontsize=14, pad=15)
>>> ax1.legend(
...     ['Mesh', '$\mathbf{\phi}$ (x-faces)', '$\partial \mathbf{phi}/\partial x$ (centers)'],
...     loc='upper right', fontsize=14
... )
>>> ax2 = fig.add_subplot(212)
>>> ax2.spy(mesh.face_x_divergence)
>>> ax2.set_title("Spy Plot", fontsize=14, pad=5)
>>> ax2.set_ylabel("Cell Index", fontsize=12)
>>> ax2.set_xlabel("X-Face Index", fontsize=12)
>>> plt.show()

(png, pdf)