I'm trying to solve a differential equation of the form
\begin{equation} \frac{dQ(t)}{dt} = - i H(t)Q(t) \end{equation}
$Q$ and $H$ are both complex matrices. I'm using RK4. I implemented it as if I was treating a single variables so I'll define $k1,k2,k3,k4$ as matrices and do the normal RK4 calculations. I'm using armadillo library with C++ so it should be able to handle complex matrices without any problem. I'm not getting the results I'm expecting so I was wondering if the Runge-Kutta should be changed for this case?
Take $L=\max_{t\in[a,b]}\|H(t)\|_{op}$, this is the global Lipschitz constant (constant of the Lipschitz condition) of the ODE over the integration interval.
Evaluating the stability region of the RK4 method gives that one needs $Lh<2.5$ for any hope of a non-chaotic result, $Lh<1.5$ for a qualitatively reasonable result, $Lh=0.1$ for a quick graphical check and $Lh\sim 0.001$ for the optimal result in double precision 64bit floating point numbers.
The topic of Runge-Kutta-like methods on manifolds like the Lie groups is a well-researched one. For example, a modification/correction of RK4 is derived and tested in Munthe-Kaas (1998): "Runge-Kutta methods on Lie groups".