.. _sphx_glr_api_gallery_ode_dynamics_integrators:

Numerical integration
---------------------

The explicit integrators behind every trajectory in this gallery: Euler's
method, Runge-Kutta, Stormer-Verlet, Dormand-Prince adaptive stepping,
Yoshida's symplectic composition, and the Adams-Bashforth multistep
methods and their stability regions.


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    <div class="sphx-glr-thumbcontainer" tooltip="Euler (1768) replaced the curve by its tangent line over a short step h: y_{n+1} = y_n + h f(t_n, y_n). Evaluating the slope at the end of the step instead gives the implicit (backward) variant y_{n+1} = y_n + h f(t_{n+1}, y_{n+1}). Both have a local error of O(h^2), which adds up to a global error of O(h): halving the step halves the error.">

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  .. image:: /api/gallery/ode_dynamics/integrators/images/thumb/sphx_glr_plot_01_euler_method_thumb.png
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  :doc:`/api/gallery/ode_dynamics/integrators/plot_01_euler_method`

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      <div class="sphx-glr-thumbnail-title">Euler's method: first-order convergence</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Runge (1895), Heun (1900) and Kutta (1901) matched more terms of the solution&#x27;s Taylor series without computing any derivatives of f. They did it by sampling the slope at several points inside each step and combining the samples. Heun&#x27;s two-stage method is second order, and Kutta&#x27;s classical four-stage method">

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  .. image:: /api/gallery/ode_dynamics/integrators/images/thumb/sphx_glr_plot_02_runge_kutta_thumb.png
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  :doc:`/api/gallery/ode_dynamics/integrators/plot_02_runge_kutta`

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      <div class="sphx-glr-thumbnail-title">Runge, Heun and Kutta: Runge-Kutta methods</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Stormer (1907) computed charged-particle orbits in the Earth&#x27;s magnetic field with the scheme later rediscovered by Verlet (1967) for molecular dynamics. For \ddot q = F(q):">

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  .. image:: /api/gallery/ode_dynamics/integrators/images/thumb/sphx_glr_plot_03_stormer_verlet_thumb.png
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  :doc:`/api/gallery/ode_dynamics/integrators/plot_03_stormer_verlet`

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      <div class="sphx-glr-thumbnail-title">Stormer-Verlet: bounded energy error over long times</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="An embedded Runge-Kutta pair computes a 5th- and a 4th-order solution from the same seven stages. Their difference estimates the local error at almost no extra cost. Dormand and Prince (1980) tuned the pair so that the 5th-order result, the one actually kept, has a very small error constant. A controller then shrinks h where the error estimate is large and grows it where it is small. On a highly eccentric Kepler orbit the step collapses at each close pass (perihelion) and stretches out far from the Sun (aphelion).">

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  .. image:: /api/gallery/ode_dynamics/integrators/images/thumb/sphx_glr_plot_04_dormand_prince_thumb.png
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  :doc:`/api/gallery/ode_dynamics/integrators/plot_04_dormand_prince`

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      <div class="sphx-glr-thumbnail-title">Dormand-Prince: embedded pairs and adaptive step size</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Yoshida (1990) raised the order of a symmetric method by composing it with itself. Three leapfrog substeps of sizes w_1 h, w_0 h, w_1 h with">

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  .. image:: /api/gallery/ode_dynamics/integrators/images/thumb/sphx_glr_plot_05_yoshida_composition_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/integrators/plot_05_yoshida_composition`

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      <div class="sphx-glr-thumbnail-title">Yoshida's symplectic composition: fourth order from leapfrog</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="John Couch Adams, computing drop shapes for Francis Bashforth&#x27;s 1883 study of capillarity, reused the slopes already computed at earlier steps instead of throwing them away. Integrating the polynomial through the last k slopes gives the order-k Adams-Bashforth formula,">

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  .. image:: /api/gallery/ode_dynamics/integrators/images/thumb/sphx_glr_plot_06_adams_bashforth_thumb.png
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  :doc:`/api/gallery/ode_dynamics/integrators/plot_06_adams_bashforth`

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      <div class="sphx-glr-thumbnail-title">Bashforth and Adams: linear multistep methods</div>
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.. toctree::
   :hidden:

   /api/gallery/ode_dynamics/integrators/plot_01_euler_method
   /api/gallery/ode_dynamics/integrators/plot_02_runge_kutta
   /api/gallery/ode_dynamics/integrators/plot_03_stormer_verlet
   /api/gallery/ode_dynamics/integrators/plot_04_dormand_prince
   /api/gallery/ode_dynamics/integrators/plot_05_yoshida_composition
   /api/gallery/ode_dynamics/integrators/plot_06_adams_bashforth