Compact Runge-Kutta Flux Reconstruction method for hyperbolic conservation laws with admissibility preservation

Abstract

Compact Runge-Kutta (cRK) Discontinuous Galerkin (DG) methods, recently introduced in [Chen, Q., Sun, Z., Xing, Y, SIAM Journal on Scientific Computing 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. The primary objective of this paper is the development of compact Runge-Kutta methods that preserve physical admissibility of the solution, such as the positivity of density and pressure. Thus, a cRK Flux Reconstruction (cRKFR) method is proposed by interpreting cRK as a procedure to approximate time-averaged fluxes. This enables the application of a flux limiter to ensure admissibility preservation in means of the scheme. The framework is a generalization of the cRKDG method and thus the admissibility preservation techniques directly apply to the original method as well. In addition, we perform the time average flux computation so that only a single numerical flux is needed for each time step which further reduces data communication in contrast to the original cRK method. Despite requiring only a single numerical flux, the time averaged numerical flux is 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. The method is further extended to handle source terms by incorporating their contributions as additional time averages. Numerical experiments involving Euler equations and the ten-moment problem are provided to validate the claims regarding the method’s accuracy, robustness, and admissibility preservation.