Runge-Kutta Algorithm

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A Runge-Kutta Algorithm is an ODE algorithm that ...



References

2016


  • (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]