基于非等距网格高阶紧致差分格式的多重网格算法研究

本文结合非等距网格高精度紧致差分格式的优越性与多重网格方法的快速收敛性，求解二维对流扩散方程。研究结果表明，对于处理物理量在不同的空间方向呈现不同的性态特征或不同变化规律的物理问题时，用非等距网格离散的四阶紧致格式的多重网格算法和二阶中心差分格式的多重网格算法都比等距网格离散得高效。同时，在非等距网格下下，部分半粗化多重网格算法比完全粗化多重网格算法具有更高的计算效率。针对不同的松弛算子对误差残量的磨光效果比较研究表明，线松弛算子是最高效的。而且，非等距网格离散的高精度紧致格式的多重网格算法对于对流扩散问题中大网格雷诺数情形也是收敛的。

The Multigrid Algorithm Based on a High-Order Compact Difference Scheme with Unequal Mesh-Size Discretization

The two-dimensional (2D) convection-diffusion equation is solved by combining the superiority of the unequal mesh-size high-order compact difference scheme with the fast convergence of the multigrid method. It is shown that for solving the physical problems in which physical quantities show different characteristic states in different directions or different changing features,the proposed multigrid algorithm based on the fourth-order compact scheme and the standard central difference scheme with unequal mesh size is more efficient than that with an equal mesh size. In addition,the partial semi-coarsening strategy is more efficient than full-coarsening with unequal mesh size discreitization. Among the multigrid algorithms of the fourth-order compact schemes,the most effective smoother is the line smoother. The multigrid algorithm of the fourth-order compact scheme under unequal mesh size discretization is convergent for the large cell Reynold number in the 2D convection-diffusion problem.