Relaxation Runge-Kutta Methods: Conservation and stability for Inner-Product Norms

We further develop a simple modification of Runge–Kutta methods that guarantees conservation or stability with respect to any inner-product norm. The modified methods can be explicit and retain the accuracy and stability properties of the unmodified Runge–Kutta method. We study the properties of the modified methods and show their effectiveness through numerical examples, including application to entropy-stability for first-order hyperbolic PDEs.