On consistency and rate of convergence of Flux Reconstruction for time-dependent problems

Abstract

This study is directed at a rigorous characterization of the consistency and convergence of discontinuous finite element schemes formulated using Flux Reconstruction (FR). We show that the FR formulation is consistent for linear advection and converges to the exact solution for any scheme that is stable in the von Neumann sense. The numerical solution for a scheme of polynomial order P is composed of P + 1 eignemodes, of which, one and exactly one is 'physical' such that it exhibits the analytical dispersion behavior in the limit of asymptotic grid resolution. The remaining P modes are 'spurious' such that the fraction of energy received by them from the initial condition vanishes in the asymptotic limit. On grid refinement, the rate of convergence of the numerical solution is a function of time, starting from a short-time rate at t = 0 + , associated with interpolation, and asymptotically approaching a long-time rate as t → ∞, associated with numerical differentiation. Both these rates can be inferred directly from the eigensystem of the numerical derivative operator. We verify these analytical expectations using simple experiments in 1-D and 2-D.

Temporal Stabilisation of Flux Reconstruction on Linear Problems

“…These are von Neumann analysis and an accompanying error analysis. We start by introducing the von Neumann analysis which will be performed in a manner similar to that introduced by Lele, 15 Huynh, 16 and Vincent et al 17 The error study aims to extend the work of Hesthaven et al 18 and Asthana et al, 19 with an extension made to show the fully-discretised temporal behaviour of harmonic solutions.…”

“…The result of Eq. (19) was first reported by Asthana and Jameson, 20 but the accompanying analysis was limited, and was followed up by Vermeire and Vincent 21 where the application of FR to implicit LES was considered. Equation (19) can be seen to be an eigenvalue problem of the update matrix, with v being the eigenmodes of the fully discretised scheme.…”

“…The dashed line is the decay time period for the primary harmonic and the other time periods are shown in dotted lines. the modes is calculated in accordance with Asthana et al:19…”

Effect of Mesh Quality on Flux Reconstruction in Multi-dimensions

“…We are concerned here with the primary mode -as FR has multiple modes, this is the one that physically represents the wave. Although, as was found by Asthana [4], this may not be how the energy distributes itself. We identify the physical mode as the mode whose dispersion relation that goes through zero and dissipation relation similar to those seen in Fig.…”

Non-Modal Analysis of Multigrid Schemes for the High-Order Flux Reconstruction Method

“…The method has been further extended by Vincent, Castonguay and Jameson [7,8] through the family of Energy-Stable Flux Reconstruction (ESFR) schemes, and it has revealed itself particularly suited for the simulation of turbulent flows using implicit Large Eddy Simulations [4,5,9,10]. Several studies about the numerical properties of FR by means of von Neumann analysis exist in the literature [11,12,13,14], mainly in the context of linear advection and advection-diffusion. Despite good experimental agreement, the von Neumann analysis has an inherent disadvantage when applied to high-order methods, since more than one eigenmode per element is obtained which forces a distinction between the primary (physical) eigenmode and the spurious ones, which are disregarded altogether.…”