High-order and energy-preserving relaxed implicit–explicit Runge–Kutta methods for Hamiltonian PDEs

This work presents a mathematical framework to develop energy-preserving, high-order linearly implicit Runge–Kutta (RK) methods for general Hamiltonian PDEs. This reformulation results in a two-stage second-order IMEX-RK method that is unconditionally energy-stable, with its diagonally implicit part satisfying the symplectic condition. In contrast, the search for a third-order IMEX-RK scheme that similarly guarantees unconditional energy-stable has been unsuccessful. To achieve a higher-order energy-preserving scheme, we incorporate a relaxation factor into the conventional IMEX-RK method, leading to the development of a class of high-order relaxed IMEX-RK (RIMEX-RK) methods. A rigorous theoretical analysis demonstrates the derived methods’ energy stability and error convergence. Key advantages of the RIMEX-RK methods include their one-step nature, high-order accuracy, energy preservation, and ease of implementation. Extensive numerical experiments demonstrate the framework’s superior performance.