In the authors' previous paper (Bai and Zhong, J. Comput. Phys. 2019), a very high-order upwind multi layer compact (MLC) scheme designed to solve smooth multi-scale flow problems with complex physics is presented. In the MLC scheme, the auxiliary equations for the first derivatives are introduced. Accordingly, the first derivatives are evolved simultaneously with the function values and used as additional degrees of freedom in the finite difference approximation. The upwind MLC scheme is subsequently derived on a centered stencil, with an adjustable parameter to introduce small dissipation for stability. It was shown in [1] that the scheme achieved spectral-like resolution within a compact stencil, and it produced very high-order accuracy in flow simulations. However, the MLC scheme shows weak numerical instabilities for a small range of wavenumbers when it is applied to multi-dimensional flows [2], which are mainly triggered by the inconsistency between its one- and two-dimensional formulations. The instability may lead to divergence in long-time multi-dimensional simulations. Moreover, the cross-derivative approximation in the MLC scheme requires an ad-hoc selection of supporting grid points. Also, the cross-derivative approximation is relatively expensive for very high-order cases. In this paper, a new directional multi-layer compact (DMLC) scheme is presented in order to address the remaining problems of the MLC scheme and obtain better performance for multi-dimensional flow simulations. The main idea of the new DMLC scheme is to introduce the auxiliary equation for the cross derivative in multi-dimensional cases, consequently, the spatial discretization can be fulfilled along each dimension independently. With this directional discretization technique, the one-dimensional formulation can be applied to all spatial derivatives in a multi-dimensional governing equation. Therefore, the DMLC scheme overcomes inconsistency between one- and two-dimensional formulations of the original MLC scheme; and it also avoids the ad-hoc cross-derivative approximations. The two-dimensional Fourier analysis demonstrates that all modes of the DMLC scheme are stable for wavenumbers in [0, 2 pi], and it has better spectral resolution and smaller anisotropic error than the MLC scheme. The stability analysis through matrix method indicates that stable boundary closure schemes are much easier to be obtained in the DMLC scheme. The numerical tests in the linear advection equation and the nonlinear Euler equations validate that the DMLC scheme are more accurate and require less CPU time than the MLC scheme on the same mesh. In particular, the long-time simulation results reveal that the DMLC scheme is always stable for both periodic and non-periodic boundary conditions; while the MLC scheme may diverge in some cases. Overall, the new DMLC scheme shows comprehensive improvements from the original MLC scheme with uniform stability, higher accuracy and spectral resolution, and better computational efficiency for multi-dimensional flows. The improvement is even more significant in very high-order cases which makes the DMLC scheme appropriate for solving complex flow problems. (C) 2019 Elsevier Ltd. All rights reserved.
