In my MSc thesis (Luan, 2007-HUS-VNU, in Vietnamese), I studied structure-preserving numerical integration for matrix differential systems whose solution is unitary (or Hermitian, skew-Hermitian, symplectic).
$M'(t) =H(M(t), t) M(t), \quad M(t_0)=M_0,$
where $M(t), M_0 \in \mathbb{C}^{n \times n}$ are differentiable unitary matrices, and $H(M(t), t)$ is a skew-Hermitian (possibly nonlinear) matrix operator. These systems often arise in decomposition techniques for matrix functions (e.g. QR or SVD). The idea was to investigate numerical methods that preserve the unitary property of the computed solution. It was verified that the Gauss-Legendre and projected Runge-Kutta methods maintain perfectly unitariness during integration and that they outperform the traditional one-step methods (RK4) as well as multistep methods such as Adam-Bashforth and BDF methods. Moreover, these schemes are also applied to compute Lyapunov exponents of a regular dynamical system.
