Very high-order multi-layer compact schemes with shock-fitting method (MLC-SF) for compressible flow simulations

High-order numerical methods are commonly employed in direct numerical simulation (DNS) to achieve the required accuracy with fewer degrees of freedom, thereby improving computational efficiency. To further improve global spatial accuracy, Bai and Zhong proposed the multi-layer compact (MLC) schemes (JCP, 2019) to introduce spatial derivatives as new degrees of freedom and create a more compact stencil for the same spatial accuracy. Stability analysis showed MLC can achieve seventh-order global accuracy with closed boundaries, which surpasses most of the sixth-order conventional upwind finite difference schemes. Despite this high-order convergence rate, MLC faces challenges in supersonic flow simulations, primarily due to the Gibbs phenomenon across shock waves. The numerical oscillation can cause divergence in high-order numerical schemes if no additional treatment, such as shock-capturing or shock-fitting methods, is applied. Therefore, further studies are needed to enhance MLC’s applicability to realistic high-speed flow applications, particularly in the context of shock treatments and boundary condition implementation. This paper develops a novel MLC method to improve its applicability for supersonic flow simulations. The proposed method integrates MLC with the shock-fitting method (MLC-SF), treating the shock wave as a computational boundary that separates upstream and downstream solutions. The shock-fitting method mitigates spurious numerical oscillations across the discontinuous interface, preserving the high-order accuracy of MLC-SF. Additionally, this paper introduces a physically consistent boundary condition for the MLC-SF spatial derivative layers behind the shock. This boundary condition uses the inversion of the flux Jacobian matrix to estimate the correct spatial derivatives, ensuring consistency between MLC-SF value and derivative layers at the inflow boundary. In order to systematically benchmark the proposed method, MLC-SF is applied to five simulation cases involving linear advection, Euler, and Navier-Stokes equations on one- and two-dimensional domains. The studied cases aim to compare the results of shock-fitting and shock-capturing methods, evaluate the performance of MLC-SF within the arbitrary Lagrangian-Eulerian (ALE) framework for moving grid applications, and test the MLC-SF derivative layers on fluid mechanics problems involving non-Cartesian grids. In both one-dimensional and two-dimensional shock wave interaction cases, MLC-SF with the proposed physically consistent inflow condition achieves seventh-order spatial accuracy, which outperforms the other four tested methods. Notably, in the one-dimensional shock-interaction results, the fifth-order WENO methods exhibit only first-order accuracy behind the shock wave, highlighting the necessity of adopting the shock-fitting approach to maintain the high spatial accuracy property in MLC-SF. In terms of computational efficiency. MLC-SF can save at least 30 % of the computational time compared to conventional high-order finite difference methods with shock-fitting for Shu-Osher-like problem. The overall objective of this study is to establish a high-order MLC framework suitable for compressible and high-speed fluid mechanics simulations.