不可压缩流动的并行数值方法

Parallel numerical methods for incompressible flows

不可压缩流动的数值模拟是计算流体力学的重要组成部分.基于有限元离散方法,本文设计了不可压缩Navier-Stokes(N-S)方程支配流的若干并行数值算法.这些并行算法可归为两大类:一类是基于两重网格离散方法,首先在粗网格上求解非线性的N-S方程,然后在细网格的子区域上并行求解线性化的残差方程,以校正粗网格的解;另一类是基于新型完全重叠型区域分解技巧,每台处理器用一局部加密的全局多尺度网格计算所负责子区域的局部有限元解.这些并行算法实现简单,通信需求少,具有良好的并行性能,能获得与标准有限元方法相同收敛阶的有限元解.理论分析和数值试验验证了并行算法的高效性.

The simulation of incompressible flows consists of an important part of computational fluid dy- namics. Based on finite element discretization, this paper develops several parallel numerical methods for the incompressible flows governed by the Navier-Stokes （N-S） equations, which can be classified into two classes： one is based on two-grid discretization, where the fully nonlinear N-S equations are first solved on a coarse grid, and the correction is then calculated on a fine grid by solving a linear residual problem in a parallel manner; the other is based on a novel fully overlapping domain decomposition technique, where each processor computes a local finite element solution in its own sub-domain using a global multiscale mesh that is locally refined around its own sub-domain. These parallel numerical methods are easy to implement. They have low communication complexity, and can yield a finite element solution with the same order of convergence rate as the standard finite element methods. Theoretical analysis and numerical tests demonstrated the high efficiency of the studied methods.