.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/ode_dynamics/integrators/plot_02_runge_kutta.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_ode_dynamics_integrators_plot_02_runge_kutta.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_ode_dynamics_integrators_plot_02_runge_kutta.py:


Runge, Heun and Kutta: Runge-Kutta methods
==========================================

Runge (1895), Heun (1900) and Kutta (1901) matched more terms of the
solution's Taylor series without computing any derivatives of :math:`f`.
They did it by sampling the slope at several points inside each step and
combining the samples. Heun's two-stage method is second order, and
Kutta's classical four-stage method

.. math::

    y_{n+1} = y_n + \tfrac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4)

is fourth order: halving :math:`h` divides the error by 16.

.. GENERATED FROM PYTHON SOURCE LINES 19-46

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.integrators import njit, rk4_integrate


    @njit
    def oscillator(state, t, params):
        return np.array([state[1], -state[0]])


    def euler_final(h, n):
        y = np.array([1.0, 0.0])
        for i in range(n):
            y = y + h * oscillator(y, i * h, None)
        return y


    def heun_final(h, n):
        y = np.array([1.0, 0.0])
        for i in range(n):
            k1 = oscillator(y, i * h, None)
            k2 = oscillator(y + h * k1, (i + 1) * h, None)
            y = y + 0.5 * h * (k1 + k2)
        return y









.. GENERATED FROM PYTHON SOURCE LINES 47-49

Error at t = 2 pi for the harmonic oscillator
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-73

.. code-block:: Python


    t_end = 2 * np.pi
    ns = 2 ** np.arange(4, 12)
    hs = t_end / ns
    exact = np.array([1.0, 0.0])
    errors = {
        "Euler (1 stage)": [np.linalg.norm(euler_final(h, n) - exact) for h, n in zip(hs, ns, strict=False)],
        "Heun (2 stages)": [np.linalg.norm(heun_final(h, n) - exact) for h, n in zip(hs, ns, strict=False)],
        "Kutta RK4 (4 stages)": [
            np.linalg.norm(rk4_integrate(oscillator, np.array([1.0, 0.0]), 0.0, h, n, np.zeros(1))[1][-1] - exact) for h, n in zip(hs, ns, strict=False)
        ],
    }

    fig, ax = plt.subplots(figsize=(7, 4.5))
    for (label, err), order in zip(errors.items(), (1, 2, 4), strict=False):
        ax.loglog(hs, err, "o-", label=label)
        print(f"{label}: error ratio for halved h = {err[-2] / err[-1]:.2f} (expected {2**order})")
    ax.set_xlabel("h")
    ax.set_ylabel("|error| after one period")
    ax.set_title("Runge-Kutta: more stages, higher order")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/ode_dynamics/integrators/images/sphx_glr_plot_02_runge_kutta_001.png
   :alt: Runge-Kutta: more stages, higher order
   :srcset: /api/gallery/ode_dynamics/integrators/images/sphx_glr_plot_02_runge_kutta_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Euler (1 stage): error ratio for halved h = 2.01 (expected 2)
    Heun (2 stages): error ratio for halved h = 4.00 (expected 4)
    Kutta RK4 (4 stages): error ratio for halved h = 16.00 (expected 16)





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

   **Total running time of the script:** (0 minutes 1.066 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_integrators_plot_02_runge_kutta.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/ode_dynamics/integrators/plot_02_runge_kutta.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_runge_kutta.ipynb <plot_02_runge_kutta.ipynb>`

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

      :download:`Download Python source code: plot_02_runge_kutta.py <plot_02_runge_kutta.py>`

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

      :download:`Download zipped: plot_02_runge_kutta.zip <plot_02_runge_kutta.zip>`


.. only:: html

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

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