One-Parameter Optimal Flux Reconstruction Schemes for Adaptive Mesh Refinement

Abstract

High-order schemes with larger time-step limits can benefit from using adaptive mesh refinement grids. One-parameter optimal flux reconstruction schemes are obtained by minimizing the error associated with wave propagation over the range of resolvable wavenumbers using a novel objective function that accounts for the Courant–Friedrichs–Lewy limit and order of accuracy constraint. The novel schemes are parameterized by the Courant–Friedrichs–Lewy limit, which, if chosen properly, leads to the obtained optimal Courant–Friedrichs–Lewy-like well-known schemes. The schemes are obtained by the differential evolution optimization algorithm, which is combined with the flux reconstruction–discontinuous Galerkin scheme to enlarge the time step limit on adaptive mesh refinement grids. Two numerical experiments were performed to investigate the properties of the schemes, including advection of a Gaussian bump and isentropic Euler vortex. The results show that the new schemes are more suitable for adaptive mesh refinement in terms of accuracy, dissipation, and stability.