I am using the XMDS library for solving the stochastic (projected) Gross-Pitaevskii equation
$$i \hbar \partial \Phi\left(\mathbf{r},t\right)_t=\hat{\mathcal{P}}\left\{(1-i \gamma)\left(\hat{H}_{\mathrm{GP}}-\mu\right) \Phi+\eta\left(\mathbf{r},t\right)\right\}$$
where $\eta\left(\mathbf{r},t\right)$ is a stochastic noise field.
In the XMDS documentation, it is noted that:
As all Runge-Kutta solutions have equal order of convergence for stochastic equations, if the step-size is limited by the stochastic term then the step-size estimation is entirely unreliable. Adaptive Runge-Kutta algorithms are therefore not appropriate for stochastic equations. (http://www.xmds.org/reference_elements.html?highlight=adaptive)
I have noted the arguments set out in this SE answer: Easily understandable argument that normal Runge–Kutta methods cannot be generalised to SDEs?.
From the XMDS documentation, they note that if the step-size $\Delta x$ is limited by $\eta$ then adaptive Runge-Kutta is unreliable. I am unsure what "limited" means in this sense... my step size is (I believe!) independent of the stochastic noise $\eta$ and is instead dependent on some circular momentum cut-off in spectral space.
Furthermore, work by A. Das et al. (Scientific Reports 2, Article number: 352 (2012)) explicitly uses adaptive Runge-Kutta methods for solving this stochastic PDE, but they do not explain why this is valid.
Why is it possible to use adaptive Runge-Kutta in this case?