两层时空加性Schwarz方法求解线性与非线性抛物方程

With the fast development of massive parallel processors computer, the "space-time" domain decomposition algorithms which are parallel both in "time" and "space" begin to receive widespread attention. This project will focus on the theoretical analysis and numerical implementation of some highly parallel two-level space-time additive Schwarz methods. The main contents include:.1. Based on the linear parabolic equation, we will construct higher order two-level space-time additive Schwarz methods, which make it possible to compute the longer period of time in the same degree of parallelism. The convergence of the new algorithms will be proved under the framework of finite element methods, and the relationship between convergence and time step and step number will be analyzed;.2. A systematic study on the nonlinear two-level space-time additive Schwarz preconditioning method for solving nonlinear parabolic equations will be presented. The equivalence between the preconditioned equation and the original problem will be proved theoretically;.3.By constructing new “space-time” restriction operators , we will propose some novel restricted space-time additive Schwarz methods, which will improve the parallelism of the algorithms further.. The project will use PETSc to implement the corresponding algorithms to verify the parallelism and numerical efficiency of the new methods. Finally, we will use the algorithms and programs that appear in the project to develop software package, so that it can be applied to solve some other time-dependent problems in parallel in both time and space dimensions.

随着超级并行机的快速发展，在“时间域”和“空间域”上同时并行计算的“时空”区域分解算法开始受到广泛关注。本项目将致力于研究高度并行的两层时空加性Schwarz方法的理论分析及其数值实现，其主要研究内容包括：.1）.以线性抛物方程为模型，构造高阶两层时空加性Schwarz方法，使得在同样的并行度内可以计算更长的时间周期。在有限元框架下证明新算法的收敛性，分析收敛性与时间步长、步数之间的关系；.2）.系统的研究非线性两层时空加性Schwarz预处理方法，实现非线性抛物方程时空并行计算。从理论上证明预处理方程与原问题的等价性；.3）.构造时空限制算子，研究限制两层时空加性Schwarz方法，进一步提高算法并行性。. 项目利用PETSc实现相应的算法，验证算法的并行性与数值有效性。最后，对所有算法与程序进行整理，开发能独立运行的程序包，以便应用于其他与时间相关问题的时空并行计算。

目前，大规模并行机器已经相当普遍，越来越多学者关注“时空并行”算法。项目主要研究内容为高度并行的两层、多层时空Schwarz方法求解线性及非线性抛物方程，在项目的支持下，得到如下研究内容：.1）.完成一系列高阶时空Schwarz方法的构造。在有限元框架下证明这些新算法的收敛性，分析收敛性与时间步长、时间窗口、空间网格大小的关系；.2）.研究了限制多层时空Schwarz方法，进一步提高算法并行性。.3）.利用PETSc实现相应的并行算法，验证了相关收敛性理论结果。通过大规模CPU核并行计算，验证算法的弱扩展性与强扩展性。.4）.从数值上验证了，只要有足够的CPU核，在大规模并行计算时间相关问题时，时空并行算法的效率要优于传统在空间并行、时间串行的并行算法。.在本项目的资助下，主要有以下成果： .(1)发表SCI文章4篇，正在审稿2篇；.(2)时空并行求解抛物方程的程序包一个，自动微分Jacobi矩阵生成器的Mathematica程序包一个；.(3)参加国内学术会议2次；培养研究生1名；.通过本项目，我们获得了一些新颖高效率的求解抛物型偏微分方程的两层水平、多层水平时空Schwarz方法。在此基础上，我们将进一步研究非线抛物问题的时空并行算法，并应用于其他重要时间相关偏微分方程问题的数值模拟。