Relaxation implicit-explicit Runge-Kutta method and its applications in highly oscillatory Hamiltonian systems

Structure-preserving numerical methods have extremely important applications in long-time simulations of highly oscillatory Hamiltonian systems. In this paper, we utilize the relaxation technique and propose a family of relaxation implicit-explicit Runge-Kutta methods for solving highly oscillatory Hamiltonian systems. Contrary to the standard implicit-explicit Runge-Kutta methods, the structure-preserving properties of the proposed methods enable them to be applied to long-time simulations. Besides, the proposed methods are linearly implicit and arbitrarily high-order accurate, which can greatly improve the computational efficiency in simulations. Finally, several numerical experiments are performed to verify the theoretical results in the article.