class brainpy.integrators.sde.ExponentialEuler(f, g, dt=None, name=None, show_code=False, var_type=None, intg_type=None, wiener_type=None, dyn_vars=None, state_delays=None)[source]#

First order, explicit exponential Euler method.

For a SDE equation of the form

\[d y=(Ay+ F(y))dt + g(y)dW(t) = f(y)dt + g(y)dW(t), \quad y(0)=y_{0}\]

its schema is given by [1]

\[\begin{split}y_{n+1} & =e^{\Delta t A}(y_{n}+ g(y_n)\Delta W_{n})+\varphi(\Delta t A) F(y_{n}) \Delta t \\ &= y_n + \Delta t \varphi(\Delta t A) f(y) + e^{\Delta t A}g(y_n)\Delta W_{n}\end{split}\]

where \(\varphi(z)=\frac{e^{z}-1}{z}\).


See also

Euler, Heun, Milstein