What numerical methods for solving differential equations are particularly well suited for solving matrix differential equations of the form
$$ \frac{dF}{dt} = A(t)F(t) ; \ \ \ F(0)= v $$
where function $F : \mathbb{R} \to \mathbb{R}^n$ is differentiable and function $A : \mathbb{R} \to \mathbb{R}^{n \times n}$ is continuous?
As mentioned in the comment there is a lot of methods for solving such system. Depending on the information on $A(t)$ you have (is it a symmetric non-negative matrix ? a symplectic one ? etc.) and the accuracy you need the answer may vary.
For example the simplest method is to replace the time derivative by $(F(t_{n+1})-F(t_n))/\Delta t$.
This leads to two different discretization
The explicit Euler method $$\frac{F(t_{n+1})-F(t_n)}{\Delta t}=A(t_n)F(t_n)$$ i.e $$F(t_{n+1})=(Id +\Delta t A(t_n) ) F(t_n).$$
The implicit Euler method $$\frac{F(t_{n+1})-F(t_n)}{\Delta t}=A(t_{n+1})F(t_{n+1})$$ i.e $F(t_{n+1})$ is solution of the system $$(Id -\Delta t A(t_{n+1}) ) F(t_{n+1})= F(t_n).$$
The explicit method is straightforward but can be unstable when the implicit one is much more stable but require to solve a linear system at each step. So depending on the context you may choose one or the other.