Abstract
In this study we employ von Neumann analyses to investigate the dispersion, dissipation, group velocity, and error properties of several fully-discrete flux reconstruction (FR) schemes. We consider three FR schemes paired with two explicit Runge–Kutta (ERK) schemes and two singly diagonally implicit RK (SDIRK) schemes. Key insights include the dependence of high-wavenumber numerical dissipation, relied upon for implicit large eddy simulation (ILES), on the choice of temporal scheme and time-step size. Also, the wavespeed characteristics of fully-discrete schemes and the relative dominance of temporal and spatial errors as a function of wavenumber and time-step size are investigated. Salient properties from the aforementioned theoretical analysis are then demonstrated in practice using linear advection test cases. Finally, a Burgers turbulence test case is used to demonstrate the importance of the temporal discretization when using FR schemes for ILES.
Highlights
Unsteady scale-resolving computational fluid dynamics (CFD) simulations, such as large eddy simulation (LES) and direct numerical simulation (DNS) rely on two different types of discretization when using the method of lines
Results from von Neumann analysis suggest that while moderately-resolved waves can achieve super-convergence, it is expected that the numerical error of well-resolved waves is limited by the order of accuracy of the temporal scheme
Another observation from von Neumann analysis is that the error in the group velocity of well-resolved waves increases with increasing time-step size
Summary
Unsteady scale-resolving computational fluid dynamics (CFD) simulations, such as large eddy simulation (LES) and direct numerical simulation (DNS) rely on two different types of discretization when using the method of lines. High-order methods, such as the FR, DG, and SD approaches, are appealing for simulations of complex unsteady flows [7,8,9,10] They have been found to provide more accurate solutions with fewer degrees of freedom and reduced computational cost relative to industry-standard second-order schemes [11]. Previous studies of the FR approach have investigated the behaviour of various energy-stable FR (ESFR) schemes using semi-discrete von Neumann analysis [6,18,19]. The current study will follow the approach of Yang et al [20] to investigate the fully-discrete behaviour of FRDG, FRSD, and a appealing ESFR scheme identified by Vermeire and Vincent [19]. We will present conclusions based on our analysis and numerical experiments, along with implications for the application of FR to large-scale ILES
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