Investigation of Shock-Capturing with Bound-Preserving Limiters for the Nonlinearly Stable Flux Reconstruction Method

TL;DR

This paper investigates the shock-capturing capabilities of a nonlinearly stable flux reconstruction scheme with bound-preserving limiters, demonstrating its advantages over standard methods in terms of robustness, accuracy, and stability for compressible flow simulations.

Contribution

It extends the bound-preserving limiter for NSFR, thoroughly tests its shock-capturing ability, and compares its performance with standard DG methods across various parameters and test cases.

Findings

NSFR scheme shows high-order accuracy and entropy stability.

Enhanced shock-capturing and oscillation mitigation over standard DG methods.

Increased robustness and larger time steps with optimized parameters.

Abstract

Nonlinearly stable flux reconstruction (NSFR) combines the key properties of provable nonlinear stability with the increased time step from energy-stable flux reconstruction. The NSFR scheme has been successfully applied to unsteady compressible flows. Through the use of a bound-preserving limiter, positivity of thermodynamic quantities is preserved, and this enables the extension of NSFR to hyperbolic conservation laws. We extend the limiter of Zhang and Shu [1] to ensure robustness for the proposed scheme. The limiter is modified to consider the minimum density and pressure at the solution nodes when determining the value to scale the solution. The modifications are thoroughly tested with a suite of test cases. In addition to these modifications, this paper conducts a thorough investigation into the shock-capturing capabilities of the NSFR scheme and the advantages it presents over standard discontinuous Galerkin (DG) methods, where, on select variants of the flux reconstruction (FR) scheme, essentially oscillation-free solutions are demonstrated. Various parameters of the scheme are extensively tested and analyzed through several 1D and 2D compressible Euler tests that verify the high-order accuracy, entropy stability, time step advantage and shock-capturing capabilities of the NSFR scheme. These parameters include the two-point flux, quadrature nodes and the strength of the FR parameter. In addition to investigating the impact of the various two-point fluxes, this paper also presents numerical studies to determine the CFL condition required to maintain positivity for the two-point flux of choice. The investigation yields insightful results for all parameters, with the results pertaining to the type of FR scheme being of special interest. The tests showcase increased robustness, time step advantages and oscillation/overshoot mitigation when employing a stronger FR parameter.