Very High-Order Upwind Multi-Layer Compact (MLC) Schemes with Spectral-Like Resolution II: Two-Dimensional Case

Abstract

In this paper, the very high-order upwind multi-layer compact (MLC) scheme developed by Bai and Zhong 1 is revisited and analyzed with focus on the two-dimensional case. The MLC scheme is designed to solve smooth multi-scale flow problems with complex physics, such as hypersonic boundary layer transition, turbulent flows, computational aeroacoustics, etc. In its multi-layer framework, the auxiliary equations for the first derivatives are introduced. Accordingly, the first derivatives are evolved simultaneously with the function values. The MLC scheme derived on structured grids has an explicit finite difference formulation, which includes both function values and first derivatives. Benefiting from the multi-layer framework, the scheme achieves very high-order accuracy and spectral-like resolution within a compact stencil. The upwind MLC scheme is derived on a centered stencil, with an adjustable parameter to introduce small dissipation for stability. A main problem of the original MLC scheme 1 is the minor numerical instability in the twodimensional case, which is mainly triggered by the inconsistency between the 1-D and 2-D MLC formulations. Also, the approximation of the cross derivative in the original scheme introduces uncertainty, and it is relatively expensive in very high-order cases. In this paper, a directional discretization technique is designed to extend the 1-D MLC scheme to twodimensional cases. By introducing the auxiliary equation for the cross derivative, the spatial discretization can be fulfilled along each dimension independently. Therefore, the 1-D MLC scheme can be applied to all spatial derivatives, and the 2-D MLC scheme is not required any more. This directional discretization technique avoids any inconsistency between the 1-D and 2-D MLC schemes, and it also overcomes the uncertainty arising from the approximation of cross derivatives. The Fourier analysis demonstrates that all modes of the new MLC scheme are stable in two-dimensional cases, and the new scheme has better spectral resolution and smaller anisotropic error for a large portion of wavenumbers in [0, 2π]. The analysis through matrix method indicates that stable boundary closure schemes are also easier to be obtained for the new MLC scheme with the directional discretization. The numerical results validate that the new MLC scheme has smaller error and better computational efficiency than the original MLC scheme due to the better spectral resolution. The long-time simulation results verify that the original MLC scheme could be unstable in some cases; while the new MLC scheme with the directional discretization is always stable for both periodic and non-periodic boundary conditions. High-Order Methods There are plenty of high-order numerical methods developed in the past few decades, such as spectral methods 2-4 , compact finite difference schemes 5-12 , discontinuous-Galerkin (DG) methods 13-16 , and nonlinear schemes like TVD 17,18 , ENO 19 , WENO 20,21. Many reviews are available for study on high-order methods. Ekaterinaris 2...