Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems
Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems
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DOI：
10.1016/j.jcp.2011.02.027
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发表时间：
2011-05
期刊：
影响因子：
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通讯作者：
Y. Ge;Fujun Cao
中科院分区：
文献类型：
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作者：
Y. Ge;Fujun Cao
本文采用Kalita等人[J.C. Kalita，A.K.张文，等.非均匀网格下对流扩散的一种无变换高阶差分格式.北京：清华大学出版社，1998.方法Fluids 44（2004）33-53]，求解二维对流扩散方程。由于HOC格式不涉及任何将非均匀网格映射到均匀网格的网格变换，因此多重网格法是求解非均匀网格上由差分方程引起的离散方程组的一种全新方法。利用面积比构造了相应的多重网格投影和插值算子。一些边界层和局部奇异性问题的应用证明了本文方法的优越性。数值结果表明，采用HOC格式的多重网格方法在非均匀网格上的收敛速度与在均匀网格上的收敛速度几乎相等，并且在非均匀网格上的计算结果保持了四阶精度，而在均匀网格上对于边界层很陡或局部奇异性很强的问题只能得到很差的解。本方法也适用于解决二维不可压Navier-Stokes方程使用流函数涡量公式和数值解的盖子驱动的空腔流动问题，并与文献中的解决方案进行了比较。
In this paper, a multigrid method based on the high order compact (HOC) difference scheme on nonuniform grids, which has been proposed by Kalita et al. [J.C. Kalita, A.K. Dass, D.C. Dalal, A transformation-free HOC scheme for steady convection–diffusion on non-uniform grids, Int. J. Numer. Methods Fluids 44 (2004) 33–53], is proposed to solve the two-dimensional (2D) convection diffusion equation. The HOC scheme is not involved in any grid transformation to map the nonuniform grids to uniform grids, consequently, the multigrid method is brand-new for solving the discrete system arising from the difference equation on nonuniform grids. The corresponding multigrid projection and interpolation operators are constructed by the area ratio. Some boundary layer and local singularity problems are used to demonstrate the superiority of the present method. Numerical results show that the multigrid method with the HOC scheme on nonuniform grids almost gets as equally efficient convergence rate as on uniform grids and the computed solution on nonuniform grids retains fourth order accuracy while on uniform grids just gets very poor solution for very steep boundary layer or high local singularity problems. The present method is also applied to solve the 2D incompressible Navier–Stokes equations using the stream function–vorticity formulation and the numerical solutions of the lid-driven cavity flow problem are obtained and compared with solutions available in the literature.