  In this paper, we investigate the energy accuracy of explicit Runge-Kutta
(RK) time discretization for antisymmetric autonomous linear systems and
present a framework for constructing RK methods with an order of energy
accuracy much greater than the number of stages. For an $s$-stage, $p$th-order
RK method, we show that the energy accuracy can achieve superconvergence with
an order up to $2s-p+1$ if $p$ is even. Several energy-superconvergent methods,
including five- to seven-stage fourth-order methods with energy accuracy up to
the eleventh order, together with their strong stability criteria, are derived.
The proposed methods are examined using several applications, including
second-order ordinary differential equations for harmonic oscillators, linear
integro-differential equations for peridynamics, and one-dimensional Maxwell's
equations of electrodynamics.
