An efficient multiscale‐like multigrid (MSLMG) method combined with a high‐order compact (HOC) difference scheme on nonuniform grids is presented to solve the two‐dimensional (2D) convection‐diffusion equations. The discrete systems with given appropriate initial solutions on two finest grids are solved to obtain the MSLMG solutions with discretization‐level accuracy by performing fewer multigrid cycles; it is implemented with alternating line Gauss–Seidel smoother, interpolation, and restriction operators on the nonuniform grids. Numerical experiments of boundary layer or local singularity problems are conducted to show that the proposed algorithm with the HOC scheme on nonuniform grids is efficient and effective to decrease the computational cost and time, and the computed approximation on the nonuniform grids has fourth order accuracy.

中文翻译：

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非均匀网格上二维对流扩散方程的高效多尺度类多网格计算

为了解决二维（2D）对流扩散方程，提出了一种有效的多尺度类多重网格（MSLMG）方法，并结合了非均匀网格上的高阶紧凑（HOC）差分方案。解决了在两个最佳网格上具有适当初始解决方案的离散系统，从而通过执行更少的多网格周期来获得具有离散化精度的MSLMG解决方案。它在非均匀网格上使用交替的高斯-赛德尔平滑器，插值和限制算子来实现。通过边界层或局部奇异性问题的数值实验表明，该算法在不均匀网格上采用HOC方案是有效且有效的降低了计算成本和时间，并且对不均匀网格的近似计算具有四阶精度。