Fully-discrete nonlinearly-stable flux reconstruction methods for compressible flows

Abstract

A fully-discrete, nonlinearly-stable flux reconstruction (FD-NSFR) scheme is developed, which ensures robustness through entropy stability in both space and time for high-order flux reconstruction schemes. We extend the entropy-stable flux reconstruction semidiscretization of Cicchino et al. [1], [2], [3] with the relaxation Runge Kutta method to construct the FD-NSFR scheme. We focus our study on entropy-stable flux reconstruction methods, which allow a larger time step size than discontinuous Galerkin. In this work, we develop an FD-NSFR scheme that prevents temporal numerical entropy change in the broken Sobolev norm if the governing equations admit a convex entropy function that can be expressed in inner-product form. For governing equations with a general convex numerical entropy function, we develop a method for implementing RRK in a flux reconstruction framework, where the semidiscrete entropy stability property is in the broken Sobolev norm. For such problems, temporal entropy change in the physical $L_2$ norm is prevented. As a result, for general convex numerical entropy, the FD-NSFR scheme achieves fully-discrete entropy stability only when the DG correction function is employed. We use entropy-conserving and entropy-stable test cases for the Burgers', Euler, and Navier-Stokes equations to demonstrate that the FD-NSFR scheme prevents temporal numerical entropy change. The FD-NSFR scheme therefore improves robustness through an entropy stability property, while the flux reconstruction filter allows for larger time steps. We find that the FD-NSFR scheme is able to recover both integrated quantities and solution contours at a higher target time-step size than the semi-discretely entropy-stable scheme, suggesting a robustness advantage for low-Mach turbulence simulations.