Energy-superconvergent Explicit Runge--Kutta Time Discretizations

This paper investigates the energy conservation properties of explicit Runge--Kutta (RK) time discretizations for autonomous skew-symmetric systems. For linear problems, we present a general framework for constructing RK methods in which the energy-accuracy order significantly exceeds the number of stages. Specifically, for an $s$-stage, $p$-th order RK method (where $p$ is even), we prove that the energy accuracy can reach up to order $2s-p+1$. Utilizing this framework, we derive several energy-superconvergent methods, including five- to seven-stage algorithms with energy accuracy up to the eleventh order, and establish their corresponding strong stability criteria. The methods are validated on a range of benchmark problems, including harmonic oscillators, integro-differential equations in peridynamics, and the Maxwell equations. Furthermore, we extend the energy-superconvergent framework to autonomous nonlinear systems with amplitude-dependent frequencies. By deriving fifth-order energy conditions for three-stage, second-order methods, we develop the RK325 algorithm. The performance of RK325 is demonstrated for a broad range of problems, including Euler's equations for rigid body dynamics, the nonlinear Schr\"odinger equation, the Korteweg--de Vries (KdV) equation, Burgers' equation, and the Landau--Lifshitz equation. Additionally, we develop four-stage, second-order methods (RK427) and five-stage, fourth-order methods (RK547), all of which achieve seventh-order energy accuracy for the cubic nonlinear case. Finally, the performance of RK547 method is illustrated using the nonlinear Maxwell--Kerr system.