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

## TL;DR

This paper introduces a modification to Runge-Kutta methods that ensures conservation or stability under any inner-product norm, maintaining original accuracy and stability, with demonstrated effectiveness in numerical examples and hyperbolic PDEs.

## Contribution

It presents a simple modification to Runge-Kutta methods that guarantees conservation or stability for any inner-product norm while preserving their original properties.

## Key findings

- Modified methods ensure stability and conservation under any inner-product norm.
- The methods retain the accuracy and stability of the original Runge-Kutta methods.
- Numerical examples demonstrate effectiveness, including in hyperbolic PDE applications.

## Abstract

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.