  A wide range of physical phenomena exhibit auxiliary admissibility criteria,
such as conservation of entropy or various energies, which arise implicitly
under the exact solution of their governing PDEs. However, standard temporal
schemes, such as classical Runge-Kutta (RK) methods, do not enforce these
constraints, leading to a loss of accuracy and stability. Previously, the
Incremental Directional Technique RK (IDT-RK) and Relaxation Runge-Kutta (R-RK)
approaches have been proposed to address this. However, these lead to a loss of
accuracy in the case of IDT-RK, or a loss of step size control in the case of
R-RK. In the current work we propose Relaxation-Free Runge- Kutta (RF-RK)
schemes, which conserve energy, maintain order of accuracy, and maintain a
constant step size, alleviating many of the limitations of the aforementioned
techniques. Importantly, they do so with minimal additional computational cost
compared to the base RK scheme. Numerical results demonstrate that these
properties are observed in practice for a range of applications. Therefore, the
proposed RF-RK framework is a promising approach for energy conservative time
integration of systems of PDEs.
