braincell.quad.heun2_step

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braincell.quad.heun2_step#

braincell.quad.heun2_step(target, *args)[source]#

Advance one step with Heun’s second-order Runge-Kutta method.

Heun’s method (also known as the explicit trapezoidal rule or improved Euler) is a two-stage, second-order explicit scheme:

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

Local truncation error is \(O(\Delta t^3)\); global error is \(O(\Delta t^2)\).

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

midpoint_step, rk2_step, ralston2_step

Notes

Butcher tableau (heun2_tableau):

\[\begin{split}\begin{array}{c|cc} 0 & 0 & 0 \\ 1 & 1 & 0 \\ \hline & \tfrac{1}{2} & \tfrac{1}{2} \end{array}\end{split}\]

Examples

>>> import brainstate
>>> import brainunit as u
>>> from braincell.quad import heun2_step
>>> with brainstate.environ.context(t=0. * u.ms, dt=0.01 * u.ms):
...     heun2_step(my_neuron)
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