RKF45#
- class brainpy.integrators.ode.RKF45(f, var_type=None, dt=None, name=None, adaptive=None, tol=None, show_code=False, state_delays=None, neutral_delays=None)[source]#
The Runge–Kutta–Fehlberg method for ODEs.
The method presented in Fehlberg’s 1969 paper has been dubbed the RKF45 method, and is a method of order \(O(h^4)\) with an error estimator of order \(O(h^5)\). The novelty of Fehlberg’s method is that it is an embedded method from the Runge–Kutta family, meaning that identical function evaluations are used in conjunction with each other to create methods of varying order and similar error constants.
Its Butcher table is:
\[\begin{split}\begin{array}{l|lllll} 0 & & & & & & \\ 1 / 4 & 1 / 4 & & & & \\ 3 / 8 & 3 / 32 & 9 / 32 & & \\ 12 / 13 & 1932 / 2197 & -7200 / 2197 & 7296 / 2197 & \\ 1 & 439 / 216 & -8 & 3680 / 513 & -845 / 4104 & & \\ 1 / 2 & -8 / 27 & 2 & -3544 / 2565 & 1859 / 4104 & -11 / 40 & \\ \hline & 16 / 135 & 0 & 6656 / 12825 & 28561 / 56430 & -9 / 50 & 2 / 55 \\ & 25 / 216 & 0 & 1408 / 2565 & 2197 / 4104 & -1 / 5 & 0 \end{array}\end{split}\]References