Assume we have a generic overdamped Langevin equation $$ \frac{d {\bf{x}} }{dt} = {\bf{f}}({\bf{x}}) + B {\bf{w}}(t) $$ where $\bf{f}$ is a deterministic external (and fixed) force field (non necessarily arising from a potential), $B>0$ is a constant and $\bf{w}$ is the usual $D$-dimensional white noise term (the position ${\bf{x}}$ is a point in $\mathbb{R}^D$).
We can simulate the dynamics by using the Euler-Maruyama method. Since $B$ is a constant, in this case the Milstein method is equivalent to the Euler-Maruyama one. Improved methods exists, for example the Runge-Kutta one, adapted to account for the noise.
I wonder if there is also something similar to an explicit Adams-Bashforth method of some order, and if there is a clear reference where explicit methods for the overdamped Langevin are discussed (I am looking at explicit methods that is accurate with the deterministic part, see the "edit" below for the specific problem).
In the case ${\bf{f}}$ is non-linear and oscillating (but bounded), is it bad practice to rely on explicit methods? Is there some "common knowledge" or "heuristics" on how to treat conveniently the integration of such an equation in $D>1$?
Edit: for definiteness, I have to deal with something like ($D=2$) $$ \frac{d { {x}} }{dt} = {{f}_1}({x,y}) + B {{w_1}}(t) \\ \frac{d { {y}} }{dt} = {{f}_2}({x,y}) + B {{w_2}}(t) $$ where $(f_1 , f_2)= (0,a) - (\partial_1 \phi , \partial_2 \phi) + r (-\partial_2 \phi , \partial_1 \phi)$. This force is not a potential one (it is only if $r=0$) and there is a constant drift term $(0,a)$. The external potential is $\phi(x,y) = \sin(x) \sin(y)$. I am interested in the average velocity, $$ \lim_{t\rightarrow \infty} \frac{{\bf{x}}(t) - {\bf{x}}(0)}{t} $$
