357 lines
18 KiB
HTML
357 lines
18 KiB
HTML
|
||
<!DOCTYPE html>
|
||
|
||
<html xmlns="http://www.w3.org/1999/xhtml">
|
||
<head>
|
||
<meta charset="utf-8" />
|
||
<title>pylib.numerical.ode module — pylib 2019.12.21 documentation</title>
|
||
<link rel="stylesheet" href="_static/alabaster.css" type="text/css" />
|
||
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
|
||
<link rel="stylesheet" href="_static/custom.css" type="text/css" />
|
||
<script type="text/javascript" id="documentation_options" data-url_root="./" src="_static/documentation_options.js"></script>
|
||
<script type="text/javascript" src="_static/jquery.js"></script>
|
||
<script type="text/javascript" src="_static/underscore.js"></script>
|
||
<script type="text/javascript" src="_static/doctools.js"></script>
|
||
<script type="text/javascript" src="_static/language_data.js"></script>
|
||
<script async="async" type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-AMS-MML_HTMLorMML"></script>
|
||
<link rel="index" title="Index" href="genindex.html" />
|
||
<link rel="search" title="Search" href="search.html" />
|
||
<link rel="next" title="pylib.numerical.ode_model module" href="pylib.numerical.ode_model.html" />
|
||
<link rel="prev" title="pylib.numerical.integration module" href="pylib.numerical.integration.html" />
|
||
|
||
<link rel="stylesheet" href="_static/custom.css" type="text/css" />
|
||
|
||
|
||
<meta name="viewport" content="width=device-width, initial-scale=0.9, maximum-scale=0.9" />
|
||
|
||
</head><body>
|
||
|
||
|
||
<div class="document">
|
||
<div class="documentwrapper">
|
||
<div class="bodywrapper">
|
||
|
||
|
||
<div class="body" role="main">
|
||
|
||
<div class="section" id="module-pylib.numerical.ode">
|
||
<span id="pylib-numerical-ode-module"></span><h1>pylib.numerical.ode module<a class="headerlink" href="#module-pylib.numerical.ode" title="Permalink to this headline">¶</a></h1>
|
||
<p>Numerical solver of ordinary differential equations.</p>
|
||
<p>Solves the initial value problem for systems of first order
|
||
ordinary differential equations.</p>
|
||
<dl class="field-list simple">
|
||
<dt class="field-odd">Date</dt>
|
||
<dd class="field-odd"><p>2015-09-21</p>
|
||
</dd>
|
||
</dl>
|
||
<span class="target" id="module-ode"></span><dl class="function">
|
||
<dt id="pylib.numerical.ode.e1">
|
||
<code class="sig-name descname">e1</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">x0</em>, <em class="sig-param">t</em>, <em class="sig-param">*p</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#e1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.e1" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Explicit first-order method /
|
||
(standard, or forward) Euler method /
|
||
Runge-Kutta 1st order method.</p>
|
||
<p>de:
|
||
Euler’sche Polygonzugverfahren / explizite Euler-Verfahren /
|
||
Euler-Cauchy-Verfahren / Euler-vorwärts-Verfahren</p>
|
||
<dl class="field-list simple">
|
||
<dt class="field-odd">Parameters</dt>
|
||
<dd class="field-odd"><ul class="simple">
|
||
<li><p><strong>f</strong> (<em>function</em>) – the function to solve</p></li>
|
||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
|
||
…)</p></li>
|
||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||
</ul>
|
||
</dd>
|
||
</dl>
|
||
<p>Approximate the solution of the initial value problem</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}\dot{x} &= f(t,x) \\
|
||
x(t_0) &= x_0\end{split}\]</div>
|
||
<p>Choose a value h for the size of every step and set</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[t_i = t_0 + i h ~,\quad i=1,2,\ldots,n\]</div>
|
||
<p>The derivative of the solution is approximated as the forward
|
||
difference equation</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\dot{x}_i = f(t_i, x_i) = \frac{x_{i+1} - x_i}{t_{i+1}-t_i}\]</div>
|
||
<p>Therefore one step <span class="math notranslate nohighlight">\(h\)</span> of the Euler method from
|
||
<span class="math notranslate nohighlight">\(t_i\)</span> to <span class="math notranslate nohighlight">\(t_{i+1}\)</span> is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}x_{i+1} &= x_i + (t_{i+1}-t_i) f(t_i, x_i) \\
|
||
x_{i+1} &= x_i + h f(t_i, x_i) \\\end{split}\]</div>
|
||
<p>Example 1:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}m\ddot{u} + d\dot{u} + ku = f(t) \\
|
||
\ddot{u} = m^{-1}(f(t) - d\dot{u} - ku) \\\end{split}\]</div>
|
||
<p>with</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\
|
||
x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\\end{split}\]</div>
|
||
<p>becomes</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}\dot{x}_1 &= x_2 \\
|
||
\dot{x}_2 &= m^{-1}(f(t) - d x_2 - k x_1) \\\end{split}\]</div>
|
||
<p>or</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}\dot{x} &= f(t,x) \\
|
||
\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &=
|
||
\begin{bmatrix} x_2 \\ m^{-1}(f(t) - d x_2 - k x_1)
|
||
\end{bmatrix} \\
|
||
&=
|
||
\begin{bmatrix} 0 \\ m^{-1} f(t) \end{bmatrix} +
|
||
\begin{bmatrix} 0 & 1 \\ -m^{-1} k & -m^{-1} d \end{bmatrix}
|
||
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]</div>
|
||
<p>Example 2:</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}m(u)\ddot{u} + d(u,\dot{u})\dot{u} + k(u)u = f(t) \\
|
||
\ddot{u} = m^{-1}(u)(f(t) - d(u,\dot{u})\dot{u} - k(u)u) \\\end{split}\]</div>
|
||
<p>with</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}x_1 &= u &\quad \dot{x}_1 = \dot{u} = x_2 \\
|
||
x_2 &= \dot{u} &\quad \dot{x}_2 = \ddot{u} \\\end{split}\]</div>
|
||
<p>becomes</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}\dot{x}_1 &= x_2 \\
|
||
\dot{x}_2 &=
|
||
m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1) \\\end{split}\]</div>
|
||
<p>or</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[\begin{split}\dot{x} &= f(t,x) \\
|
||
\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} &=
|
||
\begin{bmatrix}
|
||
x_2 \\ m^{-1}(x_1)(f(t) - d(x_1,x_2) x_2 - k(x_1) x_1)
|
||
\end{bmatrix} \\
|
||
&=
|
||
\begin{bmatrix} 0 \\ m^{-1}(x_1) f(t) \end{bmatrix} +
|
||
\begin{bmatrix}
|
||
0 & 1 \\ -m^{-1}(x_1) k(x_1) & -m^{-1} d(x_1,x_2)
|
||
\end{bmatrix}
|
||
\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\end{split}\]</div>
|
||
<p>The Euler method is a first-order method, which means that the
|
||
local error (error per step) is proportional to the square of
|
||
the step size, and the global error (error at a given time) is
|
||
proportional to the step size.</p>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="pylib.numerical.ode.e2">
|
||
<code class="sig-name descname">e2</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">x0</em>, <em class="sig-param">t</em>, <em class="sig-param">*p</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#e2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.e2" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Explicit second-order method / Runge-Kutta 2nd order method.</p>
|
||
<dl class="field-list simple">
|
||
<dt class="field-odd">Parameters</dt>
|
||
<dd class="field-odd"><ul class="simple">
|
||
<li><p><strong>f</strong> (<em>function</em>) – the function to solve</p></li>
|
||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
|
||
…)</p></li>
|
||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||
</ul>
|
||
</dd>
|
||
</dl>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="pylib.numerical.ode.e4">
|
||
<code class="sig-name descname">e4</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">x0</em>, <em class="sig-param">t</em>, <em class="sig-param">*p</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#e4"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.e4" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Explicit fourth-order method / Runge-Kutta 4th order method.</p>
|
||
<dl class="field-list simple">
|
||
<dt class="field-odd">Parameters</dt>
|
||
<dd class="field-odd"><ul class="simple">
|
||
<li><p><strong>f</strong> (<em>function</em>) – the function to solve</p></li>
|
||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
|
||
…)</p></li>
|
||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||
</ul>
|
||
</dd>
|
||
</dl>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="pylib.numerical.ode.fpi">
|
||
<code class="sig-name descname">fpi</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">xi</em>, <em class="sig-param">ti</em>, <em class="sig-param">ti1</em>, <em class="sig-param">*p</em>, <em class="sig-param">max_iterations=1000</em>, <em class="sig-param">tol=1e-09</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#fpi"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.fpi" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Fixed-point iteration.</p>
|
||
<dl class="field-list simple">
|
||
<dt class="field-odd">Parameters</dt>
|
||
<dd class="field-odd"><ul class="simple">
|
||
<li><p><strong>f</strong> (<em>function</em>) – the function to iterate <span class="math notranslate nohighlight">\(f = \dot{x}(x,t)\)</span></p></li>
|
||
<li><p><strong>xi</strong> (<em>list</em>) – initial condition <span class="math notranslate nohighlight">\(x_i\)</span></p></li>
|
||
<li><p><strong>ti</strong> (<em>float</em>) – time <span class="math notranslate nohighlight">\(t_i\)</span></p></li>
|
||
<li><p><strong>ti1</strong> (<em>float</em>) – time <span class="math notranslate nohighlight">\(t_{i+1}\)</span></p></li>
|
||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
|
||
…)</p></li>
|
||
<li><p><strong>max_iterations</strong> (<em>int</em>) – maximum number of iterations</p></li>
|
||
<li><p><strong>tol</strong> (<em>float</em>) – tolerance against residuum <span class="math notranslate nohighlight">\(\varepsilon\)</span>
|
||
(default = 1e-9)</p></li>
|
||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||
</ul>
|
||
</dd>
|
||
<dt class="field-even">Returns</dt>
|
||
<dd class="field-even"><p><span class="math notranslate nohighlight">\(x_{i}\)</span></p>
|
||
</dd>
|
||
</dl>
|
||
<div class="math notranslate nohighlight">
|
||
\[x_{i,j=0} = x_{i}\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[x_{i,j+1} = x_i + \dot{x}(x_{i,j}, t_{i+1})\cdot(t_{i+1}-t_i)\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[\text{residuum} = \frac{\lVert x_{i,j+1}-x_{i,j}\rVert}
|
||
{\lVert x_{i,j+1} \rVert} < \varepsilon\]</div>
|
||
<div class="math notranslate nohighlight">
|
||
\[x_{i} = x_{i,j=\text{end}}\]</div>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="pylib.numerical.ode.i1">
|
||
<code class="sig-name descname">i1</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">x0</em>, <em class="sig-param">t</em>, <em class="sig-param">*p</em>, <em class="sig-param">max_iterations=1000</em>, <em class="sig-param">tol=1e-09</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#i1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.i1" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Implicite first-order method / backward Euler method.</p>
|
||
<dl class="field-list simple">
|
||
<dt class="field-odd">Parameters</dt>
|
||
<dd class="field-odd"><ul class="simple">
|
||
<li><p><strong>f</strong> (<em>function</em>) – the function to solve</p></li>
|
||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
|
||
…)</p></li>
|
||
<li><p><strong>max_iterations</strong> (<em>int</em>) – maximum number of iterations</p></li>
|
||
<li><p><strong>tol</strong> (<em>float</em>) – tolerance against residuum (default = 1e-9)</p></li>
|
||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||
</ul>
|
||
</dd>
|
||
</dl>
|
||
<p>The backward Euler method has order one and is A-stable.</p>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="pylib.numerical.ode.newmark_newtonraphson">
|
||
<code class="sig-name descname">newmark_newtonraphson</code><span class="sig-paren">(</span><em class="sig-param">f</em>, <em class="sig-param">x0</em>, <em class="sig-param">xp0</em>, <em class="sig-param">xpp0</em>, <em class="sig-param">t</em>, <em class="sig-param">*p</em>, <em class="sig-param">gamma=0.5</em>, <em class="sig-param">beta=0.25</em>, <em class="sig-param">max_iterations=1000</em>, <em class="sig-param">tol=1e-09</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#newmark_newtonraphson"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.newmark_newtonraphson" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Newmark method.</p>
|
||
<dl class="field-list simple">
|
||
<dt class="field-odd">Parameters</dt>
|
||
<dd class="field-odd"><ul class="simple">
|
||
<li><p><strong>f</strong> (<em>function</em>) – the function to solve</p></li>
|
||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||
<li><p><strong>xp0</strong> (<em>list</em>) – initial condition</p></li>
|
||
<li><p><strong>xpp0</strong> (<em>list</em>) – initial condition</p></li>
|
||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
|
||
…)</p></li>
|
||
<li><p><strong>gamma</strong> (<em>float</em>) – newmark parameter for velocity (default = 0.5)</p></li>
|
||
<li><p><strong>beta</strong> (<em>float</em>) – newmark parameter for displacement (default = 0.25)</p></li>
|
||
<li><p><strong>max_iterations</strong> (<em>int</em>) – maximum number of iterations</p></li>
|
||
<li><p><strong>tol</strong> (<em>float</em>) – tolerance against residuum (default = 1e-9)</p></li>
|
||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||
</ul>
|
||
</dd>
|
||
</dl>
|
||
</dd></dl>
|
||
|
||
<dl class="function">
|
||
<dt id="pylib.numerical.ode.newmark_newtonraphson_rdk">
|
||
<code class="sig-name descname">newmark_newtonraphson_rdk</code><span class="sig-paren">(</span><em class="sig-param">fnm</em>, <em class="sig-param">x0</em>, <em class="sig-param">xp0</em>, <em class="sig-param">xpp0</em>, <em class="sig-param">t</em>, <em class="sig-param">*p</em>, <em class="sig-param">gamma=0.5</em>, <em class="sig-param">beta=0.25</em>, <em class="sig-param">max_iterations=1000</em>, <em class="sig-param">tol=1e-09</em>, <em class="sig-param">verbose=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode.html#newmark_newtonraphson_rdk"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode.newmark_newtonraphson_rdk" title="Permalink to this definition">¶</a></dt>
|
||
<dd><p>Newmark method.</p>
|
||
<dl class="field-list simple">
|
||
<dt class="field-odd">Parameters</dt>
|
||
<dd class="field-odd"><ul class="simple">
|
||
<li><p><strong>f</strong> (<em>function</em>) – the function to solve</p></li>
|
||
<li><p><strong>x0</strong> (<em>list</em>) – initial condition</p></li>
|
||
<li><p><strong>xp0</strong> (<em>list</em>) – initial condition</p></li>
|
||
<li><p><strong>xpp0</strong> (<em>list</em>) – initial condition</p></li>
|
||
<li><p><strong>t</strong> (<em>list</em>) – time</p></li>
|
||
<li><p><strong>*p</strong> – parameters of the function (thickness, diameter,
|
||
…)</p></li>
|
||
<li><p><strong>gamma</strong> (<em>float</em>) – newmark parameter for velocity (default = 0.5)</p></li>
|
||
<li><p><strong>beta</strong> (<em>float</em>) – newmark parameter for displacement (default = 0.25)</p></li>
|
||
<li><p><strong>max_iterations</strong> (<em>int</em>) – maximum number of iterations</p></li>
|
||
<li><p><strong>tol</strong> (<em>float</em>) – tolerance against residuum (default = 1e-9)</p></li>
|
||
<li><p><strong>verbose</strong> (<em>bool</em>) – print information (default = False)</p></li>
|
||
</ul>
|
||
</dd>
|
||
</dl>
|
||
</dd></dl>
|
||
|
||
</div>
|
||
|
||
|
||
</div>
|
||
|
||
</div>
|
||
</div>
|
||
<div class="sphinxsidebar" role="navigation" aria-label="main navigation">
|
||
<div class="sphinxsidebarwrapper">
|
||
<h1 class="logo"><a href="index.html">pylib</a></h1>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<h3>Navigation</h3>
|
||
<p class="caption"><span class="caption-text">Contents:</span></p>
|
||
<ul class="current">
|
||
<li class="toctree-l1 current"><a class="reference internal" href="modules.html">pylib</a><ul class="current">
|
||
<li class="toctree-l2 current"><a class="reference internal" href="pylib.html">pylib package</a></li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
|
||
<div class="relations">
|
||
<h3>Related Topics</h3>
|
||
<ul>
|
||
<li><a href="index.html">Documentation overview</a><ul>
|
||
<li><a href="modules.html">pylib</a><ul>
|
||
<li><a href="pylib.html">pylib package</a><ul>
|
||
<li><a href="pylib.numerical.html">pylib.numerical package</a><ul>
|
||
<li>Previous: <a href="pylib.numerical.integration.html" title="previous chapter">pylib.numerical.integration module</a></li>
|
||
<li>Next: <a href="pylib.numerical.ode_model.html" title="next chapter">pylib.numerical.ode_model module</a></li>
|
||
</ul></li>
|
||
</ul></li>
|
||
</ul></li>
|
||
</ul></li>
|
||
</ul>
|
||
</div>
|
||
<div id="searchbox" style="display: none" role="search">
|
||
<h3 id="searchlabel">Quick search</h3>
|
||
<div class="searchformwrapper">
|
||
<form class="search" action="search.html" method="get">
|
||
<input type="text" name="q" aria-labelledby="searchlabel" />
|
||
<input type="submit" value="Go" />
|
||
</form>
|
||
</div>
|
||
</div>
|
||
<script type="text/javascript">$('#searchbox').show(0);</script>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
</div>
|
||
</div>
|
||
<div class="clearer"></div>
|
||
</div>
|
||
<div class="footer">
|
||
©2019, Daniel Weschke.
|
||
|
||
|
|
||
Powered by <a href="http://sphinx-doc.org/">Sphinx 2.2.1</a>
|
||
& <a href="https://github.com/bitprophet/alabaster">Alabaster 0.7.12</a>
|
||
|
||
|
|
||
<a href="_sources/pylib.numerical.ode.rst.txt"
|
||
rel="nofollow">Page source</a>
|
||
</div>
|
||
|
||
|
||
|
||
|
||
</body>
|
||
</html> |