.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/pde_methods/stability/plot_02_von_neumann.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_methods_stability_plot_02_von_neumann.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_methods_stability_plot_02_von_neumann.py:


Von Neumann stability analysis
==============================

John von Neumann's method, used at Los Alamos in the 1940s and published
in 1950, inserts a single Fourier mode :math:`V_j^n = g^n e^{ij\xi}` into a
constant-coefficient scheme. The mode grows when its amplification factor
has :math:`|g(\xi)| > 1`, so stability means :math:`|g| \le 1` for every
:math:`\xi`. For the theta-scheme in log-spot,
:math:`g = (1 + (1-\theta)z)/(1 - \theta z)` with
:math:`z \approx -\sigma^2\Delta t/\Delta x^2 (1 - \cos\xi)`:
:func:`~financialkit.pde_methods.von_neumann_amplification` shows the
explicit scheme failing above a diffusion number of one, Crank-Nicolson
never growing but barely damping the highest modes, and the implicit
scheme damping them all.

.. GENERATED FROM PYTHON SOURCE LINES 19-40

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.pde_methods import von_neumann_amplification
    from financialkit.pde_methods.visualizers import plot_amplification

    sigma, r, dx = 0.2, 0.05, 0.02

    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    lams = [0.5, 0.9, 1.2]
    res = [von_neumann_amplification(sigma, r, lam * dx**2 / sigma**2, dx, theta=0.0) for lam in lams]
    plot_amplification(res, labels=[f"explicit, $\\lambda$ = {lam}" for lam in lams], ax=axes[0])
    axes[0].set_title("explicit scheme")

    big = 20 * dx**2 / sigma**2  # a diffusion number of 20
    res = [von_neumann_amplification(sigma, r, big, dx, theta=t) for t in (0.5, 0.75, 1.0)]
    plot_amplification(res, labels=["Crank-Nicolson", "theta = 0.75", "implicit"], ax=axes[1])
    axes[1].set_title(r"implicit schemes, $\lambda = 20$")
    fig.tight_layout()
    plt.show()




.. image-sg:: /api/gallery/pde_methods/stability/images/sphx_glr_plot_02_von_neumann_001.png
   :alt: explicit scheme, implicit schemes, $\lambda = 20$
   :srcset: /api/gallery/pde_methods/stability/images/sphx_glr_plot_02_von_neumann_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 41-43

The largest amplification against the diffusion number
-------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-53

.. code-block:: Python


    lam = np.linspace(0.05, 3.0, 60)
    fig, ax = plt.subplots(figsize=(6, 4))
    for theta, name in ((0.0, "explicit"), (0.5, "Crank-Nicolson"), (1.0, "implicit")):
        ax.plot(lam, [von_neumann_amplification(sigma, r, l * dx**2 / sigma**2, dx, theta=theta).max_modulus for l in lam], label=name)
    ax.axhline(1.0, color="k", lw=0.8, ls="--")
    ax.set_xlabel(r"diffusion number $\lambda = \sigma^2 \Delta t / \Delta x^2$")
    ax.set_ylabel(r"$\max_\xi |g(\xi)|$")
    ax.legend()
    plt.show()



.. image-sg:: /api/gallery/pde_methods/stability/images/sphx_glr_plot_02_von_neumann_002.png
   :alt: plot 02 von neumann
   :srcset: /api/gallery/pde_methods/stability/images/sphx_glr_plot_02_von_neumann_002.png
   :class: sphx-glr-single-img






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   **Total running time of the script:** (0 minutes 0.117 seconds)


.. _sphx_glr_download_api_gallery_pde_methods_stability_plot_02_von_neumann.py:

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      :download:`Download Python source code: plot_02_von_neumann.py <plot_02_von_neumann.py>`

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