.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde/spectral/plot_01_fourier_pseudospectral_burgers.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_01_fourier_pseudospectral_burgers.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_01_fourier_pseudospectral_burgers.py:


Fourier pseudo-spectral methods: Burgers' equation
==================================================

Steven Orszag (1969-1971) and Kreiss and Oliger (1972) made the case for
differentiating with the FFT instead of finite differences. For a
smooth periodic function the error falls faster than any power of the
grid spacing. Nonlinear terms are simply formed pointwise on the grid
("pseudo-spectral"). This script first compares Fourier and
finite-difference derivatives, then solves viscous Burgers' equation
:math:`u_t + u u_x = \nu u_{xx}` as a steepening wave, integrating the
spectral ODE system with :func:`mathematicskit.integrators.rk4_integrate`.

.. GENERATED FROM PYTHON SOURCE LINES 16-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.pde import BurgersEquation1D, fourier_derivative
    from mathematicskit.pde.visualizers import plot_snapshots








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Spectral versus finite-difference accuracy
------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-44

.. code-block:: Python


    f = lambda x: np.exp(np.sin(x))
    df = lambda x: np.cos(x) * np.exp(np.sin(x))
    ns = np.array([6, 8, 12, 16, 24, 32, 48, 64])
    spectral, central = [], []
    for n in ns:
        x = np.linspace(0, 2 * np.pi, n, endpoint=False)
        h = x[1] - x[0]
        spectral.append(np.max(np.abs(fourier_derivative(f(x)) - df(x))))
        central.append(np.max(np.abs((np.roll(f(x), -1) - np.roll(f(x), 1)) / (2 * h) - df(x))))
    fig1, ax1 = plt.subplots(figsize=(6, 4))
    ax1.semilogy(ns, np.maximum(spectral, 1e-17), "o-", label="Fourier (FFT)")
    ax1.semilogy(ns, central, "s-", label="central difference, $O(h^2)$")
    ax1.set_xlabel("grid points $N$")
    ax1.set_ylabel("max derivative error")
    ax1.set_title("Spectral accuracy: exponential convergence")
    ax1.legend()
    fig1.tight_layout()




.. image-sg:: /api/gallery/pde/spectral/images/sphx_glr_plot_01_fourier_pseudospectral_burgers_001.png
   :alt: Spectral accuracy: exponential convergence
   :srcset: /api/gallery/pde/spectral/images/sphx_glr_plot_01_fourier_pseudospectral_burgers_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 45-47

A steepening Burgers wave
-------------------------

.. GENERATED FROM PYTHON SOURCE LINES 47-57

.. code-block:: Python


    burgers = BurgersEquation1D(lambda x: np.sin(x), n=128, nu=0.02)
    sol = burgers.solve(2.0, dt=0.5 * burgers.max_stable_dt(), save_every=40)
    print(f"mean of u: initial {sol.u[0].mean():+.2e}, final {sol.final.mean():+.2e} (conserved)")
    fig2, ax2 = plt.subplots(figsize=(7, 4))
    plot_snapshots(sol, n_snapshots=6, ax=ax2)
    ax2.set_title(r"Burgers: $u_t + u u_x = 0.02\,u_{xx}$, pseudo-spectral + RK4")
    fig2.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/pde/spectral/images/sphx_glr_plot_01_fourier_pseudospectral_burgers_002.png
   :alt: Burgers: $u_t + u u_x = 0.02\,u_{xx}$, pseudo-spectral + RK4
   :srcset: /api/gallery/pde/spectral/images/sphx_glr_plot_01_fourier_pseudospectral_burgers_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    mean of u: initial -2.44e-18, final +8.67e-18 (conserved)





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

   **Total running time of the script:** (0 minutes 2.069 seconds)


.. _sphx_glr_download_api_gallery_pde_spectral_plot_01_fourier_pseudospectral_burgers.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_01_fourier_pseudospectral_burgers.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_fourier_pseudospectral_burgers.ipynb <plot_01_fourier_pseudospectral_burgers.ipynb>`

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

      :download:`Download Python source code: plot_01_fourier_pseudospectral_burgers.py <plot_01_fourier_pseudospectral_burgers.py>`

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

      :download:`Download zipped: plot_01_fourier_pseudospectral_burgers.zip <plot_01_fourier_pseudospectral_burgers.zip>`


.. only:: html

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

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