braincell.quad.rk3_step

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braincell.quad.rk3_step#

braincell.quad.rk3_step(target, *args)[source]#

Advance one step with the classical third-order Runge-Kutta method.

The classical (Kutta’s) three-stage RK3 scheme is

\[\begin{split}k_1 &= f(t_n, y_n), \\ k_2 &= f\!\left(t_n + \tfrac{1}{2}\Delta t,\ y_n + \tfrac{1}{2}\Delta t \, k_1\right), \\ k_3 &= f\!\left(t_n + \Delta t,\ y_n - \Delta t \, k_1 + 2 \Delta t \, k_2\right), \\ y_{n+1} &= y_n + \tfrac{\Delta t}{6}\left(k_1 + 4 k_2 + k_3\right).\end{split}\]

Local truncation error is \(O(\Delta t^4)\); global error is \(O(\Delta t^3)\).

Parameters:

  • target (DiffEqModule) – Differential-equation module to advance.

  • *args – Extra positional arguments forwarded to target’s integration hooks.

Returns:

Updates target’s state in place.

Return type:

None

See also

heun3_step, ralston3_step, ssprk3_step

rk4_step

Fourth-order classical Runge-Kutta.

Notes

Butcher tableau (rk3_tableau):

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

Examples

>>> import brainstate
>>> import brainunit as u
>>> from braincell.quad import rk3_step
>>> with brainstate.environ.context(t=0. * u.ms, dt=0.01 * u.ms):
...     rk3_step(my_neuron)
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