utils.solver.heun_maruyama_method

utils.solver.heun_maruyama_method(
    f,
    t_span,
    y0,
    dt,
    si=None,
    noise_func=None,
    rng=None,
    args=(),
    post_step=None,
)

Fixed-step Heun-Maruyama integrator for stochastic differential equations.

A predictor-corrector scheme that achieves strong order 1.0 for SDEs with additive noise (diffusion independent of state) and weak order 2.0. This is a meaningful improvement over :func:euler_maruyama_method (strong order 0.5) when transition timing is the quantity of interest, as in bistable climate models.

Solves:

dy = f(t, y) dt + g(t, y) dW

using the two-stage Heun scheme::

ỹ  = y  + f(t, y) dt + g(t, y) dW          (Euler predictor)
y' = y  + ½[f(t, y) + f(t+dt, ỹ)] dt
        + ½[g(t, y) + g(t+dt, ỹ)] dW       (Heun corrector)

A single Wiener increment dW is shared between predictor and corrector steps, which is the standard approach for strong convergence.

Parameters

f : callable

Drift function with signature f(t, y, *args).

t_span : tuple of float

(t0, tf) integration bounds.

y0 : array - like

Initial state vector.

dt : float

Integration timestep.

si : float or None = None

Sampling interval — output is saved every si time units, with Wiener increments accumulated across the intervening sub-steps (no interpolation). Must be an integer multiple of dt. Defaults to dt (every step is saved).

noise_func : callable or None = None

Diffusion function with signature noise_func(t, y), returning a vector of per-state diffusion scales. If None, the stochastic term is zero and the Heun deterministic ODE solver is recovered.

rng : numpy.random.Generator or None = None

Random generator for Wiener increments. A fresh generator is created if None.

args : tuple = ()

Extra positional arguments forwarded to f.

post_step : callable or None = None

Optional hook called after each accepted sub-step with signature post_step(t, y) -> y. The returned array replaces the current state, allowing post-step corrections (e.g. state nudging).

Returns

solution : Solution

Object with attributes t (time axis) and y (state trajectory).

Raises

: ValueError

If t_span is invalid, si is not an integer multiple of dt, t_span length is not an integer multiple of si, or noise_func returns a vector whose shape does not match the state vector.

Notes

For purely additive noise (g constant or state-independent) this scheme achieves strong order 1.0. For multiplicative noise (g depends on y) strong order drops back toward 0.5 but weak order 2.0 is retained, still outperforming Euler-Maruyama in distribution-level statistics. See Rößler (2010) for a full convergence analysis.