View source for Runge-Kutta Algorithm

A [[Runge-Kutta Algorithm]] is an [[ODE algorithm]] that ...
* <B>See:</B> [[Temporal Discretization]], [[Ordinary Differential Equation]], [[Numerical Ordinary Differential Equations]], [[Butcher Tableau]].

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== References ==

=== 2016 ===
* (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Runge–Kutta_methods Retrieved:2016-1-11.
** In [[numerical analysis]], the '''Runge–Kutta methods</B> are a family of implicit and explicit iterative methods used in [[temporal discretization]] for the approximate solutions of [[ordinary differential equation]]s. These methods were developed around 1900 by the German mathematicians [[Carl David Tolmé Runge|C. Runge]] and [[Martin Wilhelm Kutta|M. W. Kutta]].         <P>        See the article on [[numerical ordinary differential equations|numerical methods for ordinary differential equation]]s for more background and other methods. See also [[List of Runge–Kutta methods]].
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* (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods Retrieved:2016-1-11.
** <B>[[Runge–Kutta methods]]</B> are methods for the numerical solution of the [[ordinary differential equation]] : <math> \frac{d y}{d t} = f(t, y)\, </math> which take the form : <math> y_{n+1} = y_n + h \sum_{i=1}^s b_i k_i\, </math> : <math> k_i = f\left(t_n + c_i h, y_n + h \sum_{j = 1}^{s} a_{ij} k_j\right). </math> The methods listed on this page are each defined by its [[Butcher tableau]], which puts the coefficients of the method in a table as follows: : <math> \begin{array}{c|cccc} c_1 & a_{11} & a_{12}& \dots & a_{1s}\\ c_2 & a_{21} & a_{22}& \dots & a_{2s}\\ \vdots & \vdots & \vdots& \ddots& \vdots\\ c_s & a_{s1} & a_{s2}& \dots & a_{ss} \\ \hline & b_1 & b_2 & \dots & b_s\\ \end{array} </math>

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