随机Navier-Stokes方程空间-时间并行算法的研究

The project considers the parallel computing for stochastic Navier-Stokes flow field effected by a number of random factors and their coupling. The random factors are illustrated expansion by multivariate polynomial expansion method, and the coupled equations are mainly given in the form of products generated by polynomial noise associated with random variables by using Galerkin method. In order to avoid double counting process, Our approach starts with a careful analysis of the sequence of systems, orders them appropriately, and then puts them into separate groups. Then, we introduce and study and parallel space-time multigrid method. we develop some overlapping Schwarz methods whose subdomains cover both space and time variables, and we show that the methods work well for stochastic Navier-Stokes equations discretized with an implicit stochastic Galerkin method. One- and two-level Schwarz preconditioned recycling GMRES methods are carefully investigated such that many components of the methods are reused when a large number of linear systems are solved. The key elements of this approach include an ordering algorithm and two grouping algorithms. We present some experimental results obtained on a parallel computer with more than one thousand processors. The successful completion of the project will bring significant new developments in parallel computing for random fluid. The successful completion of the project will bring significant new developments in parallel computing random fluid.

本项目将考虑多个随机因素及其耦合作用影响的随机Navier-Stokes流场的并行计算问题. 对影响流场的随机因素利用多元多项式展开，通过Galerkin方法利用与随机变量相关的多项式噪音内积得到耦合的方程组. 为避免方程组求解过程中的重复计算， 将对方程组定义排序算法，将方程组解耦分组. 接下来，引进和研究高紧致格式空间-时间分裂并行算法，高紧致差分离散每组方程，空间－时间分裂，构建可回收Krylov子空间，在每个Krylov子空间开发重叠的Schwarz预处理技术. 因Krylov子空间既包括空间又包括时间变量，深度研究一级和二级 Schwarz 预处理回收GMRES方法， 使得该预处理方法能用于被解决了的大量的线性系统. 最终数值实验演示该算法的适用性和并行性能. 该项目的顺利完成将为随机流体的并行计算带来新的重要进展.

本项目将考虑多个随机因素及其耦合作用影响的随机Navier-Stokes流场的并行计算问题. 对影响流场的随机因素利用多元多项式展开，通过Galerkin方法利用与随机变量相关的多项式噪音内积得到耦合的方程组. 为避免方程组求解过程中的重复计算， 将对方程组定义排序算法，将方程组解耦分组. 接下来，引进和研究高紧致格式空间-时间分裂并行算法，高紧致差分离散每组方程，空间－时间分裂，构建可回收Krylov子空间，在每个Krylov子空间开发重叠的Schwarz预处理技术. 因Krylov子空间既包括空间又包括时间变量，深度研究一级和二级 Schwarz 预处理回收GMRES方法， 使得该预处理方法能用于被解决了的大量的线性系统. 最终数值实验演示该算法的适用性和并行性能. 该项目的顺利完成将为随机流体的并行计算带来新的重要进展.