Temporal Stabilisation of Flux Reconstruction on Linear Problems

Abstract

Filtering is often used in Large Eddy Simulation with a global filter width, instead here a filter width in the reference domain of high order Flux Reconstruction is considered. It is shown via Von Neumann analysis how filtering effects the dispersion and dissipation of the scheme when spatially and temporally discretised. With it being shown that filtering stabilises the scheme temporally by upto 25% for forth order FR. The impact of filtering on error production is calculated, highlighting the reduction in convective velocity caused and showing numerically the impact on order of accuracy. Finally, the turbulent Taylor-Green case is used to understand the effect of reference domain filtering on the transition to turbulence, and a filter Reynolds number is defined that is shown to be useful in understanding the effect of filtering on simulations.

On Fourier analysis of polynomial multigrid for arbitrary multi-stage cycles

“…The additional prolongation steps, in both cases, causes a greater redistribution of energy from the primary mode to the secondary mode. When considered with the knowledge that the secondary modes have shorter half-lives [23], the mechanism of convergence acceleration is understood to come from this redistribution. The additional restriction smoothing steps in the n s = 3 case diminishes the redistribution, and hence is why the contraction factor in Figure 4.4b show poorer acceleration.…”

A New Family of Weighted One-Parameter Flux Reconstruction Schemes

“…The exact details of the definition of Q are not included as this is secondary to the aim of the paper, however details can be found in several works [3,12,27,28]. Then defining the diagonalisation of Q as:…”

“…To calculate the rate of convergence, we need to monitor the semi-discrete error of the solution for different grid spacings. The derivation of the analytical semi-discrete and fully-discrete error can be found from Asthana et al [27] and Trojak et al [28] and, as the derivation is secondary to the aim, here we will jump to the end result for the semi-discrete error.…”

“…There are two rates of convergence that can be considered, the initial rate as t → 0 and the long time rate as t → ∞, with the initial rate being dictated by the secondary modes and the long time rate by the primary mode. Due to the secondary modes having very short half-lives at a well-resolved wavenumbers [28], we consider the most important convergence rate to be r h (t) → ∞, and this is shown in Figure 3. It is evident then that for both the central difference and upwinded interface case improvement may be made to the rate of convergence over standard DG schemes.…”

Research on high-order flux reconstruction method with adaptive mesh for shock capturing

“…In artificial viscosity, the LAD (Localized Artificial Diffusivity) [23] and LLAV (Localized Laplacian Artificial Viscosity) [24] and have been applied widely [25,26,27,28]. And there are some other ways of shock capturing like filtering technique can be found in [29]…”