.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/spectral/plot_02_chebyshev_collocation.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_pde_spectral_plot_02_chebyshev_collocation.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_pde_spectral_plot_02_chebyshev_collocation.py:


Chebyshev collocation: spectral accuracy without periodicity
============================================================

Fourier methods need periodic problems. Gottlieb and Orszag's 1977
monograph set out the theory of the non-periodic alternative: collocate
at the Chebyshev points :math:`x_k = -\cos(k\pi/N)`. These points cluster
at the ends and so avoid the Runge phenomenon. This script solves the
boundary-value problem :math:`u'' = f` with Chebyshev collocation and
with second-order finite differences, and watches the Chebyshev error
fall geometrically while the finite-difference error only falls like
:math:`N^{-2}`. It closes with a 2D Poisson problem on a square.

.. GENERATED FROM PYTHON SOURCE LINES 16-31

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import chebyshev_poisson_1d, chebyshev_poisson_2d, solve_poisson_1d
    from mathematicskit.pde.visualizers import plot_field_2d


    def exact(x):
        return np.exp(x) * np.sin(3 * x)


    def source(x):
        return np.exp(x) * (-8 * np.sin(3 * x) + 6 * np.cos(3 * x))









.. GENERATED FROM PYTHON SOURCE LINES 32-34

Geometric versus algebraic convergence
--------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-52

.. code-block:: Python


    ns = np.arange(6, 41, 2)
    cheb_err, fd_err = [], []
    for n in ns:
        c = chebyshev_poisson_1d(source, n=n, ua=exact(-1.0), ub=exact(1.0))
        d = solve_poisson_1d(source, -1.0, 1.0, n, exact(-1.0), exact(1.0))
        cheb_err.append(np.max(np.abs(c.u - exact(c.x))))
        fd_err.append(np.max(np.abs(d.u - exact(d.x))))
    fig1, ax1 = plt.subplots(figsize=(6, 4))
    ax1.semilogy(ns, np.maximum(cheb_err, 1e-17), "o-", label="Chebyshev collocation")
    ax1.semilogy(ns, fd_err, "s-", label="finite differences")
    ax1.set_xlabel("grid points $N$")
    ax1.set_ylabel("max error")
    ax1.set_title(r"$u'' = f$ on $[-1, 1]$")
    ax1.legend()
    fig1.tight_layout()
    print(f"N = 20: Chebyshev error {cheb_err[7]:.1e}, finite-difference error {fd_err[7]:.1e}")




.. image-sg:: /api/gallery/pde/spectral/images/sphx_glr_plot_02_chebyshev_collocation_001.png
   :alt: $u'' = f$ on $[-1, 1]$
   :srcset: /api/gallery/pde/spectral/images/sphx_glr_plot_02_chebyshev_collocation_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    N = 20: Chebyshev error 6.7e-15, finite-difference error 1.6e-02




.. GENERATED FROM PYTHON SOURCE LINES 53-55

The points cluster at the boundary
----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 55-65

.. code-block:: Python


    c = chebyshev_poisson_1d(source, n=16, ua=exact(-1.0), ub=exact(1.0))
    fig2, ax2 = plt.subplots(figsize=(7, 3.5))
    x = np.linspace(-1, 1, 400)
    ax2.plot(x, exact(x), color="black", label="exact")
    ax2.plot(c.x, c.u, "o", color="tab:red", label="16 Chebyshev points")
    ax2.set_xlabel("$x$")
    ax2.legend()
    fig2.tight_layout()




.. image-sg:: /api/gallery/pde/spectral/images/sphx_glr_plot_02_chebyshev_collocation_002.png
   :alt: plot 02 chebyshev collocation
   :srcset: /api/gallery/pde/spectral/images/sphx_glr_plot_02_chebyshev_collocation_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 66-68

A 2D Poisson problem on a square
--------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 68-76

.. code-block:: Python


    sol = chebyshev_poisson_2d(lambda X, Y: 10 * np.sin(8 * X * (Y - 1)), n=24)
    fig3, ax3 = plt.subplots(figsize=(5.5, 4.5))
    plot_field_2d(sol, ax=ax3, levels=25)
    ax3.set_title(r"$\nabla^2 u = 10 \sin(8x(y-1))$, $u = 0$ on the boundary")
    fig3.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/pde/spectral/images/sphx_glr_plot_02_chebyshev_collocation_003.png
   :alt: $\nabla^2 u = 10 \sin(8x(y-1))$, $u = 0$ on the boundary
   :srcset: /api/gallery/pde/spectral/images/sphx_glr_plot_02_chebyshev_collocation_003.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.430 seconds)


.. _sphx_glr_download_api_gallery_pde_spectral_plot_02_chebyshev_collocation.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/pde/spectral/plot_02_chebyshev_collocation.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_chebyshev_collocation.ipynb <plot_02_chebyshev_collocation.ipynb>`

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

      :download:`Download Python source code: plot_02_chebyshev_collocation.py <plot_02_chebyshev_collocation.py>`

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

      :download:`Download zipped: plot_02_chebyshev_collocation.zip <plot_02_chebyshev_collocation.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_