Compact Runge-Kutta flux reconstruction methods with entropy and/or kinetic energy preserving fluxes
This work addresses robustness issues in numerical simulations of hyperbolic equations, but it is incremental as it builds on existing methods by integrating known fluxes into a specific framework.
The authors tackled the problem of enhancing robustness in high-order methods for hyperbolic conservation laws by incorporating entropy or kinetic energy preserving fluxes into the Compact Runge-Kutta flux reconstruction framework, and they observed improved robustness in numerical experiments for compressible Euler, MHD, and multi-ion MHD equations.
Compact Runge-Kutta (cRK) methods are a class of high order methods for solving hyperbolic conservation laws characterized by their compact stencil including only immediate neighboring finite elements. A Compact Runge-Kutta flux reconstruction (cRKFR) method for solver hyperbolic conservation laws was introduced in [Babbar, A., Chen, Q., Journal of Scientific Computing, 2025] which uses a time average flux formulation to perform evolution using a single numerical flux computation at each step, making it a single stage method. Entropy or kinetic energy preserving numerical fluxes are often used for construction of high order entropy stable or kinetic energy preserving methods for hyperbolic conservation laws, and are known to enhance the robustness of numerical methods for under-resolved simulations. In this work, we show how these fluxes can be incorporated into the cRKFR framework for general hyperbolic equations that consist of fluxes and non-conservative products. We test the effectiveness of this new class of methods through numerical experiments for the compressible Euler equations, magnetohydronamics (MHD) equations and multi-ion MHD equations. It is observed that the application of entropy or kinetic energy preserving fluxes enhances the robustness of the cRKFR methods.