  Using a recent characterization of energy-preserving B-series, we derive the
explicit conditions on the coefficients of a Runge-Kutta method that ensure
energy preservation (for Hamiltonian systems) up to a given order in the step
size, which we refer to as the pseudo-energy-preserving (PEP) order. We study
explicit Runge-Kutta methods with PEP order higher than their classical order.
We provide examples of such methods up to PEP order six, and test them on
Hamiltonian ODE and PDE systems. We find that these methods behave similarly to
exactly energy-conservative methods over moderate time intervals and exhibit
significantly smaller errors, relative to other Runge-Kutta methods of the same
order, for moderately long-time simulations.
