# Below, we demonstrate 1) how to apply the face-y divergence operator,
# and 2) the mapping of the face-y 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 :math:`\boldsymbol{\phi}` defined on the
# y-faces, we take the y-derivative by constructing the face-y 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_y = mesh.faces_y
phi = np.exp(-(faces_y[:, 1] ** 2) / 8** 2)
#
# Construct the y-divergence operator and apply to vector
#
Dy = mesh.face_y_divergence
dphi_dy = Dy @ phi
#
# Plot original function and the y-divergence
#
fig = plt.figure(figsize=(13, 6))
ax1 = fig.add_subplot(121)
w = np.r_[np.ones(mesh.nFx), phi]  # Need vector on all faces for image plot
mesh.plot_image(w, ax=ax1, v_type="Fy")
ax1.set_title("Scalar on y-faces", fontsize=14)
ax2 = fig.add_subplot(122)
mesh.plot_image(dphi_dy, ax=ax2)
ax2.set_yticks([])
ax2.set_ylabel("")
ax2.set_title("Y-derivative at cell center", fontsize=14)
plt.show()
#
# The discrete y-face divergence operator is a sparse matrix that maps
# from y-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 :math:`\boldsymbol{\phi}` and its
# y-derivative :math:`\partial \boldsymbol{\phi}/ \partial y` 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_y[:, 0], mesh.faces_y[:, 1], "g^", markersize=8
)
for ii, loc in zip(range(mesh.nFy), mesh.faces_y):
    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-Y Divergence", fontsize=14, pad=15)
ax1.legend(
    ['Mesh',r'$\mathbf{\phi}$ (y-faces)',r'$\partial_y \mathbf{\phi}/\partial y$ (centers)'],
    loc='upper right', fontsize=14
)
ax2 = fig.add_subplot(212)
ax2.spy(mesh.face_y_divergence)
ax2.set_title("Spy Plot", fontsize=14, pad=5)
ax2.set_ylabel("Cell Index", fontsize=12)
ax2.set_xlabel("Y-Face Index", fontsize=12)
plt.show()