# Adaptive Runge-Kutta for Stochastic (Projected) Gross-Pitaevskii Equation

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?

## 1 Answer

It turns out that adaptive step-size methods are appropriate for this equation given it is a Langevin equation in the limit of weak, additive noise. [1]

It should be noted (if anyone were to stumble upon this post), that stochastic differential equations with strong multiplicative noise are typically unstable and not appropriate for adaptive step-size methods.

[1] See G. N. Milstein and M. V. Tretyakov's general discussion of SDEs in Stochastic Numerics for Mathematical Physics, Springer-Verlag Berlin Heidelberg, pp. 211-283, (2004).