Length-scales for efficient CFL conditions in high-order methods with distorted meshes: Application to local-timestepping for p-multigrid

We propose a strategy to estimate the maximum stable time-steps for explicit time-stepping methods for hyperbolic systems in a high-order flux reconstruction framework. The strategy is derived through a von-Neumann analysis (VNA) framework for the advection-diffusion equation on skewed two- and three-dimensional meshes. It directly incorporates the spatial polynomial- and mesh-discretization in estimating the convective and diffusive length-scales. The strategy is extended to the density-based Navier-Stokes system of equations, taking into account the omnidirectionality of the speed of sound.

We compare the performance of this strategy with three other popular choices of length-scales across a wide range of polynomial-orders, meshes of drastically varying cell-quality, and flow-physics. The proposed strategy shows robust behavior across all test-scenarios with limited variation of the maximum stable CFL-number (0.1 to 1) for polynomial-orders 1 through 10,
unlike other strategies where the CFL-number varies sharply. Finally, we show the advantage of the proposed methodology for local-timestepping for p-multigrid through a RANS-modeled steady-state turbulent flow case, on a mesh with large disparity of mesh elements and aspect ratios.