Fully-Discretely 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, 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.

Figures (23)

• Figure 1: Convergence plots for the inviscid Burgers' test case using $c_{DG}$. The solid lines use SD-NSFR, and the dashed lines use FD-NSFR. Left: convergence of $L^2$ solution error at the end time compared to a calculation using a very small time step. Center: convergence of the average relaxation parameter to $1$ in the FD-NSFR cases. Right: energy change in the broken Sobolev $W^{k,2}_c$ norm at the end time relative to the initial condition. Machine zero is shown as $10^{-17}$ such that it can be plotted on a logarithmic scale.
• Figure 2: Error convergence for Burgers' equation using various FR correction parameters. The solid lines use SD-NSFR, and the dashed lines use FD-NSFR. Left: $L^2$ norm of error at the end of the simulation, compared to a calculation using a very small time step. Center: convergence of the average relaxation parameter to $1$ in the FD-NSFR cases. Right: Energy change in the broken Sobolev $W^{k,2}_c$ norm at the end of the calculation. Machine zero is shown as $10^{-17}$ such that it can be plotted on a logarithmic scale.
• Figure 3: Evolution of energy change using the FD-NSFR (left) and SD-NSFR (center) schemes and $\Delta t = 0.005$. The norms are evaluated in the broken Sobolev norm. Right: Evolution of relaxation parameter $\gamma^n$ using the fully-discrete scheme and $\Delta t = 0.005$.
• Figure 4: Evolution of energy in the $L^2$ (top) and $W^{k,2}_c$ (bottom) norms for the Burgers' test case using FD-NSFR. Energy is conserved exactly in the $W^{k,2}_c$ norm, but is only conserved in the $L^2$ norm for $c_{DG}$, where $\mathcal{K}=\mathbf{0}$.
• Figure 5: Energy dissipation in the viscous Burgers' test case. The bottom figure subtracts the normalized energy of the reference solution from the indicated large-time-step solution. The left figure shows the relaxation parameter evolution for the FD-NSFR case.

Theorems & Definitions (14)

• Remark 1
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 2
• Remark 3
• Lemma 1
• Remark 4
• Lemma 2