RK3#

class brainpy.integrators.ode.RK3(f, var_type=None, dt=None, name=None, show_code=False, state_delays=None, neutral_delays=None)[source]#

Classical third-order Runge-Kutta method for ODEs.

For the given initial value problem \(y'(x) = f(t,y);\, y(t_0) = y_0\), the third order Runge-Kutta method is given by:

\[y_{n+1} = y_n + 1/6 ( k_1 + 4 k_2 + k_3),\]

where

\[\begin{split}k_1 = h f(t_n, y_n), \\ k_2 = h f(t_n + h / 2, y_n + k_1 / 2), \\ k_3 = h f(t_n + h, y_n - k_1 + 2 k_2 ),\end{split}\]

where \(t_n = t_0 + n h.\)

Error term \(O(h^4)\), correct up to the third order term in Taylor series expansion.

The Taylor series expansion is \(y(t+h)=y(t)+\frac{k}{6}+\frac{2 k_{2}}{3}+\frac{k_{3}}{6}+O\left(h^{4}\right)\).

The corresponding Butcher tableau is:

\[\begin{split}\begin{array}{c|ccc} 0 & 0 & 0 & 0 \\ 1 / 2 & 1 / 2 & 0 & 0 \\ 1 & -1 & 2 & 0 \\ \hline & 1 / 6 & 2 / 3 & 1 / 6 \end{array}\end{split}\]