New very high-order upwind multi-layer compact (MLC) schemes with spectral-like resolution for flow simulations

Abstract

Numerical simulations of multi-scale flow problems such as hypersonic boundary layer transition, turbulent flows, computational aeroacoustics and various other flow problems with complex physics require high-order methods with high spectral resolutions. For instance, the receptivity mechanisms in the hypersonic boundary layer are the resonant interactions between forcing waves and boundary-layer waves, and the complex wave interactions are difficult to be accurately predicted by conventional low-order numerical methods. High-order methods, which are robust and accurate in resolving a wide range of time and length scales, are required. Currently, the high-order finite difference methods for simulations of hypersonic flows are usually upwind schemes or compact schemes with fifth-order accuracy or lower [1]. The objective of this paper is to develop and analyze a new very high-order numerical scheme with the spectral-like resolution for flow simulations on structured grids, with focus on smooth flow problems involving multiple scales. Specifically, a new upwind multi-layer compact (MLC) scheme with spectral-like resolution up to seventh order is derived in a finite difference framework. By using the ‘multi-layer’ idea, which introduces first derivatives into the MLC schemes and approximates the second derivatives, the resolution of the MLC schemes can be significantly improved within a compact grid stencil. The auxiliary equations are required and they are the only nontrivial equations, which contributes to good computational efficiency. In addition, the upwind MLC schemes are derived based on the idea of constructing upwind schemes on centered stencils with adjustable parameters to control the dissipation. Fourier analysis is performed to show that the MLC schemes have small dissipation and dispersion in a very wide range of wavenumbers in both one- and two-dimensional cases, and the anisotropic error is much smaller than conventional finite difference methods in the two-dimensional case. Furthermore, the stability analysis with matrix method shows that high-order boundary closure schemes are stable because of compactness of the stencils. The accuracies and rates of convergence of the new schemes are validated by numerical experiments of the linear advection equation, the nonlinear Euler equations, and the Navier–Stokes equations. The numerical results show that good computational efficiency, very high-order accuracies, and high spectral resolutions especially on coarse meshes can be attained with the MLC schemes. Overall, the MLC scheme has the properties of simple formulations, high-order accuracies, spectral-like resolutions, and compact stencils, and it is suitable for accurate simulation of smooth multi-scale flows with complex physics.