.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples_gallery/fields/finite_differences.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_gallery_fields_finite_differences.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_gallery_fields_finite_differences.py:


Finite differences approximation
================================

This example displays various finite difference (FD) approximations of derivatives of
simple harmonic function.

.. GENERATED FROM PYTHON SOURCE LINES 8-69



.. image-sg:: /examples_gallery/fields/images/sphx_glr_finite_differences_001.png
   :alt: First derivative of $f(x) = \sin(x)$, Second derivative of $f(x) = \sin(x)$, First derivative of $g(x) = \sin(x)^2$, Second derivative of $g(x) = \sin(x)^2$
   :srcset: /examples_gallery/fields/images/sphx_glr_finite_differences_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from pde import CartesianGrid, ScalarField
    from pde.tools.expressions import evaluate

    # create grids with different resolution to emphasize finite difference approximation
    grid_fine = CartesianGrid([(0, 2 * np.pi)], 256, periodic=True)
    grid_coarse = CartesianGrid([(0, 2 * np.pi)], 10, periodic=True)

    # create figure to present plots of the derivative
    fig, axes = plt.subplots(nrows=2, ncols=2, sharex=True, sharey=True)

    # plot first derivatives of sin(x)
    f = ScalarField.from_expression(grid_coarse, "sin(x)")
    f_grad = f.gradient("periodic")  # first derivative (from gradient vector field)
    ScalarField.from_expression(grid_fine, "cos(x)").plot(
        ax=axes[0, 0], label="Expected f'"
    )
    f_grad.plot(ax=axes[0, 0], label="FD grad(f)", ls="", marker="o")
    plt.legend(frameon=True)
    plt.ylabel("")
    plt.xlabel("")
    plt.title(r"First derivative of $f(x) = \sin(x)$")

    # plot second derivatives of sin(x)
    f_laplace = f.laplace("periodic")  # second derivative
    f_grad2 = f_grad.divergence("periodic")  # second derivative using composition
    ScalarField.from_expression(grid_fine, "-sin(x)").plot(
        ax=axes[0, 1], label="Expected f''"
    )
    f_laplace.plot(ax=axes[0, 1], label="FD laplace(f)", ls="", marker="o")
    f_grad2.plot(ax=axes[0, 1], label="FD div(grad(f))", ls="", marker="o")
    plt.legend(frameon=True)
    plt.xlabel("")
    plt.title(r"Second derivative of $f(x) = \sin(x)$")

    # plot first derivatives of sin(x)**2
    g_fine = ScalarField.from_expression(grid_fine, "sin(x)**2")
    g = g_fine.interpolate_to_grid(grid_coarse)
    expected = evaluate("2 * cos(x) * sin(x)", {"g": g_fine})
    fd_1 = evaluate("d_dx(g)", {"g": g})  # first derivative (from directional derivative)
    expected.plot(ax=axes[1, 0], label="Expected g'")
    fd_1.plot(ax=axes[1, 0], label="FD grad(g)", ls="", marker="o")
    plt.legend(frameon=True)
    plt.title(r"First derivative of $g(x) = \sin(x)^2$")

    # plot second derivatives of sin(x)**2
    expected = evaluate("2 * cos(2 * x)", {"g": g_fine})
    fd_2 = evaluate("d2_dx2(g)", {"g": g})  # second derivative
    fd_11 = evaluate("d_dx(d_dx(g))", {"g": g})  # composition of first derivatives
    expected.plot(ax=axes[1, 1], label="Expected g''")
    fd_2.plot(ax=axes[1, 1], label="FD laplace(g)", ls="", marker="o")
    fd_11.plot(ax=axes[1, 1], label="FD div(grad(g))", ls="", marker="o")
    plt.legend(frameon=True)
    plt.title(r"Second derivative of $g(x) = \sin(x)^2$")

    # finalize plot
    plt.tight_layout()
    plt.show()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 5.055 seconds)


.. _sphx_glr_download_examples_gallery_fields_finite_differences.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: finite_differences.ipynb <finite_differences.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: finite_differences.py <finite_differences.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: finite_differences.zip <finite_differences.zip>`