braincell.quad.ralston2_step

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braincell.quad.ralston2_step#

braincell.quad.ralston2_step(target, *args)[source]#

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

Ralston’s second-order method is the two-stage explicit Runge-Kutta scheme that minimises the leading-order truncation error coefficient among second-order RK2 variants:

\[\begin{split}k_1 &= f(t_n, y_n), \\ k_2 &= f\!\left(t_n + \tfrac{2}{3}\Delta t,\ y_n + \tfrac{2}{3}\Delta t \, k_1\right), \\ y_{n+1} &= y_n + \tfrac{\Delta t}{4}\left(k_1 + 3 k_2\right).\end{split}\]

It is therefore identical to rk2_step() and useful when an error-optimal RK2 scheme is desired without committing to higher cost. 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

rk2_step, midpoint_step, heun2_step

Notes

Butcher tableau (ralston2_tableau):

\[\begin{split}\begin{array}{c|cc} 0 & 0 & 0 \\ \tfrac{2}{3} & \tfrac{2}{3} & 0 \\ \hline & \tfrac{1}{4} & \tfrac{3}{4} \end{array}\end{split}\]

References

Examples

>>> import brainstate
>>> import brainunit as u
>>> from braincell.quad import ralston2_step
>>> with brainstate.environ.context(t=0. * u.ms, dt=0.01 * u.ms):
...     ralston2_step(my_neuron)
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