| 摘要 | 该文在理论上的主要贡献就是对于一般线性椭圆型问题,提出了一种确定其最优（近优）Robin参数的极其简洁的方法.中科院软件所并行中心研制的并行油藏模拟器PRIS_1.0已成功地运行于各种并行环境下,而且其线性解法器的性能已达到国际先进水平.但PRIS_1.0的解法器是基于矩阵的,没有考虑原来的偏微分方程组的性质.我们首先研究了椭圆型偏微分方程的最优（近优）Robin参数的确定方法（最优Robin参数携带了微分方程的信息,因而可据此大大改善经加法Schwarz预处理后矩阵问题的性质）,然后,为了方便应用,我们把最优Robin参数的确定方法推广到由椭圆型偏微分方程离散化得到的矩阵问题.但是由于时间限制和程序实现的复杂性,我们仅仅在油藏模拟器的压力方程求解中实现了Robin参数的应用.该文从两个方面探讨了这个问题：●在传统的加法Schwarz方法的基础上,对内边界做特殊处理.●在内边界上采用Dirichlet传递条件是造成传统的加法Schwarz方法收敛缓慢的主要原因,所以我们可以采用更为一般的传递条件-Robin传递条件. |
| 其他摘要 | Domain Decomposition Methods originated from classical Schwarz Alternate Methods(1870). After several decade's development, especially with the flourishing of parallel computing in recent years, the related theories for Domain Decomposition Methods have become more and more mature. Whereas, some problems have not been settled and need further efforts to fix them. For elliptic problems, the convergence rate of Overlapping Schwarz methods depends on how to transport the boundary information quickly from one side to other sides. The classical overlapping Schwarz methods try to transport boundary information by exchanging message on the artificial boundaries. However because of taking Dirichlet conditions as the transmission conditions, which makes the solution reflect when across the artificial boundaries, classical overlapping Schwarz methods converge very slowly-suppose the overlapping domains take a little percentage. Robin transmission conditions can remedy this problem to a large extent. However, till now, we lack a effective approach to determine the proper Robin parameters for general elliptic problems. And the main theoretical contributions of this dissertation are: for general linear elliptic problems, we developed a very simple way to determine the optimal or nearly optimal Robin parameters. RDCPS(R k D Center for Parallel Software, Institute of Software, Chinese Academy of Sciences) had developed a parallel reservoir simulator named PRIS-1.0. and it has running on all kinds of parallel platforms. However, the linear solver in PRIS-1.0 has not involved any properties of the corresponding partial differential equations, which makes the solver less effective as the problem scale and the number of subdomains get more large. Under the background of 973 project (National Key Basis Research Project, No. G1999032803) and the guiding of Prof. Sun, m this dissertation, we try to take advantage of the properties of the underlying partial differential equations in parallel reservoir simulation software. In this dissertation, two different ways will be addressed to accelerate the rimvergenee rate of overlapping Schwarz methods. Keep the transmission conditions posed on the artificial boundaries unchanged, and just insert some other steps into the iteration process of classical overlap ping Schwarz methods, which should deal with the artificial boundaries or coarse grid. Take other more accurate boundary conditions as the transmission conditions, which would reduce the reflection greatly. So the exchanging of information between subdomains should be speeded up greatly. In this way, the transporta tion of boundary information would also be promoted greatly. The main contributions of this dissertation are In chapter 2, from the viewpoint of distributed memory and parallel computing, a new insight into the overlapping Schwarz methods will be obtained by checking how the overlapping domains be dealt with, which would help us to unify all kinds of overlapping Schwarz methods and Bblock Jacobi method. In chapter 4 and chapter 5, the Dirichlet transmission conditions for overlap ping Schwarz methods will be replaced by Robin transmission conditions when applied to general linear elliptic partial differential equations. And a very simple approach to determine the optimal or nearly optimal robin parameters will be developed. And numerical experiments showed that , for model prob lems the Robin parameters can improve the convergence rate of overlapping Schwarz methods by more than 10 times if the scale of the underlying problems are moderate large. In order to apply the robin transmission conditions to some practical matrix problems in a straight way, the above approach will be generalized to matrix problems which result from the discretization of elliptical partial differential equations. And as generalization, the matrixes can be non-elliptical on some points. In chapter 7, we will analyze a large scale sequential oil reservoir simulator. Subsequently some parts of it will be parallelized. And then the Robin parameters will be applied to the pressure matrix solver which plays an important role in the whole matrix solver. Finally, numerical experiments for the data (grids = 33 x 44 x 35) provided by DaQing oil field will be implemented on ^^ clusters. And numerical experiments suggested that Overlapping is an useful technique to improve the convergence rate and preconditioning efficiency of the Schwarz methods. For our pressure equation solver, compared to nonover-lapping Block Jacobi method, the overlapping additive Schwarz methods can improve the efficiency of the whole solver by 30% if the the overlapping per centage be kept under 10%(for both iteration steps and iteration time) On the other hand, if the overlapping percentage exceeds 10%, then because the subproblems are too big, the iteration time will increase though the iteration steps will decrease further. |
修改评论