Analysis of Recovery-assisted discontinuous Galerkin methods for the compressible Navier-Stokes equations

Abstract

The discontinuous Galerkin (DG) method has been recognized as a promising approach for high-fidelity numerical simulations of turbulent flows (e.g., large-eddy simulation or direct numerical simulation), given its capability to achieve high-order accuracy in complex geometries while damping out spurious high-wavenumber oscillations in advection-dominated regimes. In this work, we analyze an advection-diffusion scheme arising from the strategic use of the Recovery concept to improve the performance of the basic DG spatial discretization on a nearest-neighbors stencil. Following the principle of Recovery, the underlying solution between two neighboring cells is reconstructed in an interface-centered fashion to compute the convective and diffusive fluxes with exceptional accuracy. While certain Recovery-based schemes are known to achieve exceedingly high convergence rates for diffusion (up to 3p+2 in the cell-average norm, where p is the degree of the polynomial basis), there are more constraining order of accuracy limitations for advection due to the direction of the wind. The key technical challenge we address is to develop an advection-diffusion scheme that is both stable and efficient while leveraging the high accuracy of the Recovery concept on a nearest-neighbors stencil. The usage of Recovery overcomes a fundamental deficiency in typical DG discretizations, namely that a conventional advection-diffusion scheme is reduced from order 2p+1 accuracy in the advection-dominated regime to order 2p accuracy in the diffusion-dominated regime, as demonstrated by Fourier analysis. Our new Recovery-assisted scheme is instead able to achieve order 2p+2 accuracy in both the advection-dominated and diffusion-dominated regimes on the nearest-neighbors stencil. By combining the Recovery operator with the mixed formulation, the proposed advection-diffusion scheme achieves improved accuracy compared to established mixed formulation approaches while circumventing the differentiation of the recovered solution, which is a liability in multi-dimensional geometries. Fourier analysis and a suite of linear and nonlinear test problems, including 3D compressible Navier-Stokes, are presented to examine the performance of the new Recovery-assisted scheme; the results show a considerable accuracy advantage compared to a conventional, state-of-the-art DG approach. Furthermore, to simplify the implementation, we show that the Recovery procedure can be recast as a set of derivative-based correction terms, which replicates the Recovery operator on structured meshes while avoiding the traditional complexities of the Recovery operation.