Compact Runge-Kutta Flux Reconstruction for Hyperbolic Conservation Laws with admissibility preservation

TL;DR

This paper introduces a compact Runge-Kutta Flux Reconstruction method for hyperbolic conservation laws that reduces data communication, maintains optimal accuracy, and preserves admissibility, even for nonsmooth solutions.

Contribution

It proposes a novel cRK flux reconstruction scheme with a single flux computation per step, combined with limiters for admissibility and nonsmooth solutions, extending to source terms.

Findings

Achieves optimal accuracy across all polynomial degrees.

Maintains CFL conditions similar to cRKDG methods.

Successfully preserves admissibility and positivity in numerical tests.

Abstract

Compact Runge-Kutta (cRK) Discontinuous Galerkin (DG) methods, recently introduced in [Q. Chen, Z. Sun, and Y. Xing, SIAM J. Sci. Comput. SIAM J. Sci. Comput., 46: A1327-A1351, 2024], are a variant of RKDG methods for solving hyperbolic conservation laws and are characterized by their compact stencil including only immediate neighboring finite elements. This article proposes a cRK Flux Reconstruction (FR) method by interpreting cRK as a procedure to approximate time-averaged fluxes, which requires computing only a single numerical flux for each time step and further reduces data communication. The numerical flux is carefully constructed to maintain the same Courant-Friedrichs-Lewy (CFL) numbers as cRKDG methods and achieve optimal accuracy uniformly across all polynomial degrees, even for problems with sonic points. A subcell-based blending limiter is then applied for problems with nonsmooth solutions, which uses Gauss-Legendre solution points and performs MUSCL-Hancock reconstruction on subcells to mitigate the additional dissipation errors. Additionally, to achieve a fully admissibility-preserving cRKFR scheme, a flux limiter is applied to the time-averaged numerical flux to ensure admissibility preservation in the means, combined with a positivity-preserving scaling limiter. The method is further extended to handle source terms by incorporating their contributions as additional time averages. Numerical experiments including Euler equations and the ten-moment problem are provided to validate the claims regarding the method's accuracy, robustness, and admissibility preservation.