Relaxation Runge-Kutta Methods: Conservation and stability for Inner-Product Norms
Provides a general framework for ensuring conservation/stability in numerical ODE solvers, relevant to computational scientists and engineers.
The paper develops a modification of Runge-Kutta methods that ensures conservation or stability for any inner-product norm, preserving accuracy and stability of the original methods, demonstrated through numerical examples including entropy-stable PDEs.
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.