An Adaptive Flux Reconstruction Scheme for Robust Shock Capturing

TL;DR

This work develops Adaptive Flux Reconstruction (AFR), an adaptive extension of the NSFR framework that combines the accuracy of DG in smooth regions with the oscillation-damping of FR near shocks. AFR uses a Persson modal shock sensor to scale a surface lifting operator via a parameter $c$ in $[0,c_+]$, yielding $c_{ ext{sensor}}=oldsymbol{c} imes c_+$ and ensuring $0\le \varepsilon\le 1$ for $c_{ ext{sensor}}$. The scheme preserves local and global conservation and remains discretely entropy stable, provided a suitable entropy flux is used; a positivity-preserving limiter and a flexible sensor allow robust, essentially oscillation-free solutions. Numerical results across smooth Gaussian pulses and challenging shock-dominated problems (Leblanc shock tube, shock diffraction, and double Mach reflection) demonstrate higher CFL capabilities, reduced oscillations, and accurate capture of key flow features compared with DG and FR baselines, highlighting AFR’s practical impact for robust high-order shock-capturing.

Abstract

In the case of hyperbolic conservation laws, high-order methods, such as the classical DG method, experience the phenomenon of unwanted high-frequency oscillations in the vicinity of a shock. Shock-capturing methods such as artificial dissipation, solution, flux, or TVD limiting are generally used to eliminate non-physical oscillations and provide bounds on physical quantities. For entropy-stable schemes, the additional objective would be to retain provable entropy dissipation guarantees of the underlying scheme, i.e. subcell limiting or entropy filtering [1, 2, 3, 4]. The nonlinearly-stable flux reconstruction (NSFR) semi-discretization given in Eq. 7 with a suitable flux reconstruction scheme has been demonstrated to mitigate spurious oscillations in the presence of shock discontinuities and at CFL values substantially larger than the DG variant of the NSFR scheme whilst retaining the property of entropy stability [5]. NSFR schemes achieve this by introducing an alternative lifting operator for surface numerical flux penalization, albeit at the expense of accuracy. In this technical note, we present an adaptive approach to the choice of the lifting operator employed, which maintains higher accuracy and allows for larger CFL values while retaining the underlying provable attributes of the scheme. While it cannot eliminate oscillations such as the aforementioned shock-capturing methods, together with a positivity-preserving limiter, the scheme provides for solutions that are essentially oscillation-free.

Figures (6)

• Figure 1: [Gaussian Pulse] Plot of orders of convergence for the test results using DG, adaptive and FR schemes. Top row (p2): $L_1$ (Left), $L_2$ (Middle) and $L_\infty$ (Right). Bottom row (p3): $L_1$ (Left), $L_2$ (Middle) and $L_\infty$ (Right).
• Figure 2: [Leblanc Shock Tube] Plots of density and pressure within $15 \leq x \leq 18.75$ for $p=3$, NSFR-Ra, Roe dissipation for the $c_{DG}$, $c_+$ and AFR schemes with the plots underneath indicating the value of $c$ used for the AFR scheme
• Figure 3: [Shock Diffraction] P3 solution at $t=2.3s$ using Adaptive scheme - Density (left) and FR Parameter (right)
• Figure 4: [Shock Diffraction] Plot of density across $(1,0)\rightarrow(13,11)$ at $t=2.3s$ using for all three schemes (left) with a closer look at the wake of the shock (right)
• Figure 5: [DMR] P3 Density solution at $t=0.2s$, using schemes - DG (top), Adaptive (middle), FR (bottom)

Theorems & Definitions (6)

• Remark 1
• Definition 1
• Theorem 1
• proof
• Remark 2
• Remark 3