基于高阶耗散紧致格式的GMRES方法收敛特性研究

计算效率较低是当前限制高阶精度计算方法应用的重要因素。为了提高高阶精度混合型耗散紧致格式（HDCS）的计算效率，发展了适合多块对接网格的广义最小残值（GMRES）方法，并利用GMRES方法开展了HDCS格式的加速收敛研究。首先研究了GMRES的预处理方法、CFL数和内层迭代步数对HDCS数值模拟收敛特性的影响，计算结果显示：点松弛方法是一种高效的预处理方法；CFL数对计算收敛速度影响较大；GMRES方法存在最优的内层迭代步数。利用GMRES方法完成了NACA 0012翼型绕流、NLR 7301翼型绕流和DLR-F4翼身组合体绕流的数值模拟，并与其他隐式时间推进方法进行了对比，GMRES方法计算更加稳定，并且计算效率相对LU-SGS（Lower-Upper Symmetric Gauss-Seidel）方法可以提高5倍以上。研究结果表明，本文发展的GMRES方法在多块对接网格中具有良好的计算稳定性，计算结果的残差可以收敛到更低的量级，并且可以较大幅度地提高高阶精度数值模拟的计算效率。

Convergence Property Investigation of GMRES Method Based on High-order Dissipative Compact Scheme

Low computational efficiency is an important factor constraining the application of high-order numerical methods. To improve the computational efficiency of hybrid cell-edge and cell-node dissipative compact scheme (HDCS), a generalized minimum residual (GMRES) algorithm suitable for multi-block structured grids is developed to accelerate simulations. The influence of GMRES's precondition methods, CFL number and sub-iteration number on convergence property of HDCS high-order simulations is investigated. It is shown that the point relaxation method is an efficient precondition method, that the CFL number can greatly affect the computational efficiency, and that GMRES has an optimal sub-iteration number. GMRES is applied to simulations of NACA 0012 airfoil, NLR 7301 airfoil and DLR-F4 wing/body configuration, and is compared with other implicit time integration methods. By using GMRES, the computation becomes more stable, and the computational efficiency can be improved by more than 5 times when compared with the LU-SGS(Lower-Upper Symmetric Gauss-Seidel) method. The results indicate that the GMRES method developed in this paper has good stability in multi-block structured grids, the residual can converge to lower levels, and GMRES can greatly improve the computational efficiency of high-order simulations.