Compact Runge-Kutta flux reconstruction methods with entropy and/or kinetic energy preserving fluxes

Abstract

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.

Figures (4)

• Figure 1: Convergence test for the cRKFR scheme for (a) the isentropic vortex for the compressible Euler equations (with KEP flux from kennedygruber2008) and (b) manufactured solution for the multi-ion MHD equations (with the EC flux from ramirez2025).
• Figure 2: Kelvin-Helmholtz instability for the compressible Euler equations using the cRKFR scheme with the KEP flux of kennedygruber2008 and the blending scheme of babbar2024admissibilitybabbar2025crk with $\alpha_\text{max} = 0.008$. The density profile is shown at different times using polynomial degree $N=3$ and $64 \times 64$ elements.
• Figure 3: Multi-ion MHD Kelvin-Helmholtz instability using the EC flux of ramirez2025 without the subcell-based blending limiter. The density profile of ion species 1 is shown at different times using polynomial degree $N=3$ and $64 \times 128$ elements.
• Figure 4: Richtmyer-Meshkov instability for the compressible Euler equations using the cRKFR scheme with the KEP flux of kennedygruber2008 and blending scheme of babbar2024admissibilitybabbar2025crk with $\alpha_\text{max} = 0.001$. The density profile is shown at different times using polynomial degree $N=3$ and $32 \times 96$ elements.