Fully-Discretely Nonlinearly-Stable Flux Reconstruction Methods for Compressible Flows

TL;DR

This paper introduces a fully-discrete, nonlinearly-stable flux reconstruction scheme for compressible flows that enhances robustness and allows larger time steps while maintaining entropy stability, demonstrated on Burgers', Euler, and Navier-Stokes equations.

Contribution

It develops a novel FD-NSFR scheme that ensures entropy stability in both space and time, extending previous methods with relaxation Runge Kutta and addressing temporal entropy change.

Findings

Enables larger time steps compared to semi-discrete schemes.

Prevents temporal numerical entropy change in key equations.

Shows robustness in low-Mach turbulence simulations.

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, 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 allows for a larger time step size while maintaining the robustness offered by entropy-stable schemes. 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.