Spatial eigensolution analysis of discontinuous Galerkin schemes with practical insights for under-resolved computations and implicit LES

Abstract

a b s t r a c tThe study focusses on the dispersion and diffusion characteristics of discontinuous spectral element methods -specifically discontinuous Galerkin (DG) -via the spatial eigensolution analysis framework built around a one-dimensional linear problem, namely the linear advection equation. Dispersion and diffusion characteristics are of critical importance when dealing with under-resolved computations, as they affect both the numerical stability of the simulation and the solution accuracy. The spatial eigensolution analysis carried out in this paper complements previous analyses based on the temporal approach, which are more commonly found in the literature. While the latter assumes periodic boundary conditions, the spatial approach assumes inflow/outflow type boundary conditions and is therefore better suited for the investigation of open flows typical of aerodynamic problems, including transitional and fully turbulent flows and aeroacoustics. The influence of spurious/reflected eigenmodes is assessed with regard to the presence of upwind dissipation, naturally present in DG methods. This provides insights into the accuracy and robustness of these schemes for under-resolved computations, including under-resolved direct numerical simulation (uDNS) and implicit large-eddy simulation (iLES). The results estimated from the spatial eigensolution analysis are verified using the one-dimensional linear advection equation and successively by performing two-dimensional compressible Euler simulations that mimic (spatially developing) grid turbulence.

On the performance of a high-order multiscale DG approach to LES at increasing Reynolds number

“…Recent research has highlighted the deficiencies of the LLF scheme for the simulation of low-Mach number turbulence in the context of under-resolved DG simulations of the Euler equations [54,55] and in studies of 2D grid turbulence [53], based on a no-model, or implicit LES (ILES), approach.…”

Non-modal analysis of spectral element methods: Towards accurate and robust large-eddy simulations

“…For numerical schemes with more than one degree of freedom (DOF) per computational cell, such as in highorder SEM, several ways of investigating the diffusion characteristics of the scheme are possible. The most widely used technique is the eigensolution analysis; which has been succesfully applied to CG [41], standard DG [24,25,38,40], hybridized DG [44] and FR [39] methods. Eigenanalyses address the diffusion and dispersion characteristics, in wavenumber space, of the discretization of linear propagation-type problems, such as the linear convection or convection-diffusion equation in one dimension.…”

A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations

“…Spectral element methods combine the advantageous properties of finite element/finite volume and spectral methods, namely geometric flexibility and reduced dispersion/dissipation errors (see e.g. [21,59,62] on the latter topic). Nevertheless, the application of spectral element methods to challenging problems such as turbulent flows over complex geometries is still somewhat hindered by numerical instability issues.…”