I would like to solve a system of second-order differential equations to describe the dynamics of a system of particles.Two Newton-like forces are responsible for the motion of each particle $i$: A force acting on each particle due to other particles $f_{i,j}$ and a stochastic term of noise $f_{\mathrm{noise}}$.The force acting on each particle due to other particles $f_{i,j}$ depends on the current position $s_i$ and velocity $v_i$ of particle $i$ and the position $s_j$ and velocity $v_j$ of the other particles $j$ of the system. $$F_i= f_{i,j}(v_i,v_j,s_i,s_j) + f_{\mathrm{noise}}$$  The two components of motion in 2D are included for every term before mentioned.Under an Euler scheme, the velocity and position of each particle would be updated as follows:$$\begin{alignat}{1}v_i & v_{i} + \frac{F_i}{m} \Delta t \\s_i & s_i + v_i \Delta t\end{alignat}$$where $m$ is the mass of particle $i$ and $\Delta t$ is the integration step.However, I would like to use the Milsteins algorithm for the velocity update (since we have a term of noise) and the fourth-order Runge-Kutta method to update the position $s_i$. I am confused due to having $f_{i,j}$ dependent on $s_i$, $s_j$, $v_i$ and $v_j$. How should I operate?
Runge Kutta and Milstein  system of second-order coupled differential equations with noise
runge kutta;stochastic;differential equations;particle