Runge-Kutta Algorithm

See: Temporal Discretization, Ordinary Differential Equation, Numerical Ordinary Differential Equations, Butcher Tableau.

References

(Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Runge–Kutta_methods Retrieved:2016-1-11.
- In numerical analysis, the Runge–Kutta methods are a family of implicit and explicit iterative methods used in temporal discretization for the approximate solutions of ordinary differential equations. These methods were developed around 1900 by the German mathematicians C. Runge and M. W. Kutta.
  See the article on numerical methods for ordinary differential equations for more background and other methods. See also List of Runge–Kutta methods.

(Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods Retrieved:2016-1-11.
- Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation : [math]\displaystyle{ \frac{d y}{d t} = f(t, y)\, }[/math] which take the form : [math]\displaystyle{ y_{n+1} = y_n + h \sum_{i=1}^s b_i k_i\, }[/math] : [math]\displaystyle{ 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]\displaystyle{ \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]