High-order methods for computational fluid dynamics: A brief review of compact differential formulations on unstructured grids

Abstract

Popular high-order schemes with compact stencils for Computational Fluid Dynamics (CFD) include Discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV) methods. The recently proposed Flux Reconstruction (FR) approach or Correction Procedure using Reconstruction (CPR) is based on a differential formulation and provides a unifying framework for these high-order schemes. Here we present a brief review of recent developments for the FR/CPR schemes as well as some pacing items. IntroductionIn the field of Computational Fluid Dynamics (CFD), low -order methods are generally robust and reliable; as a result, they are routinely employed in practical calculations. For the same computing cost, high-order methods can provide considerably more accurate solutions, but they are more complicated and less robust. The need to improve and develop new high-order methods with favorable properties has attracted the interest of many researchers as evidenced by the recently held First (2012) and Second (2013) International Workshops on High-Order CFD Methods.The Discontinuous Galerkin (DG) method is currently among the most widely used high-order numerical methods for solving the compressible Navier-Stokes equations on unstructured meshes. It was introduced for the neutron transport equation by Reed and Hill (1973), analyzed by LaSaint and Raviart (1974) and developed and made popular for fluid dynamics equations by Cockburn,

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

“…The present study investigates the dispersion and diffusion/dissipation properties of discontinuous Galerkin (DG) spectral element methods, see e.g. [14,[27][28][29] , through the spatial eigensolution analysis framework applied to the one-dimensional linear advection equation. The findings obtained through the eigensolution analysis are successively verified in linear advection simulations and via a two-dimensional test case that resembles the behaviour of grid turbulence at very high-Reynolds numbers.…”

High-order computational fluid dynamics tools for aircraft design

“…Note that equation (3.15) can be written as and other members of the family were discovered in [54,55]. See [42] for a comprehensive review of recent developments.…”

“…In order to accurately compute noise for the entire flight envelope, accurate near-field aerodynamic simulations call for high-order methods. Petrov-Galerkin) [27,28], staggered grid multi-domain [29], high-order k-exact finite volume [30,31], residual distribution [32], DG [33][34][35][36], spectral volume/difference [37][38][39], PnPm [40] and correction procedure via reconstruction (CPR) [41][42][43]. Which high-order methods will be embraced by future design tools?…”

Assessment of high-order finite volume methods on unstructured meshes for RANS solutions of aeronautical configurations

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