An Adaptive Flux Reconstruction Scheme for Robust Shock Capturing

The nonlinearly stable flux reconstruction (NSFR) method is an attractive implementation that combines the key properties of provable nonlinear stability and the increased time-step from energy stable flux reconstruction (ESFR) schemes. NSFR recovers various high-order methods through the use of correction functions expressed in the form of filtered discontinuous Galerkin (DG) methods. This paper presents a novel adaptive flux reconstruction (AFR) method for hyperbolic conservation laws wherein the choice of lifting operator is implemented adaptively on a cell-by-cell basis. For problems involving strong shocks, the adaptive implementation uses the DG scheme across the domain except cells where shocks or discontinuities are detected. These cells employ the FR scheme. This allows for a larger time step to be used while maintaining a high order of accuracy. The AFR framework incorporates a modal discontinuity sensor to control the adaptation of the FR parameter and a bound-preserving limiter to preserve positivity of thermodynamic quantities. The method is tested with a suite of 1D and 2D Euler test cases. The 2D Gaussian Pulse convergence test is performed to demonstrate that AFR retains a high order of accuracy with a lower magnitude of error than the original FR scheme. Following this, the 1D BlastWave and Leblanc Shock Tube cases demonstrate the robustness advantages of the new scheme compared to the classical DG scheme. Furthermore, the Double Mach Reflection case is run to demonstrate the shock-capturing capabilities of the scheme.