Mathematical Sciences: Domain Decomposition Algorithms for Linear Hyperbolic Equations

在流体动力学、跨音速流动、水库模拟、天气预报和结构分析等领域，使用并行计算机求解偏微分方程是很重要的，在这些领域，许多感兴趣的问题没有超级计算机就无法解决。应用并行计算机来解决这些问题的一种技术被称为域分解，其中感兴趣的域被细分为几个较小的子域，并且在每个子域问题上求解偏微分方程的任务被分配给不同的处理器。然后将在各个子域上计算的解拼凑在一起，得到全局解。目前在应用区域分解技术方面的大部分工作都是在椭圆型偏微分方程领域，而对双曲型方程的关注很少。研究者提出用区域分解的方法求解线性双曲型方程。在线性双曲方程的背景下，引入了使用重叠域的思想来开发一种适用于并行处理器的域分解算法。提出了通信成本和负载均衡问题。研究者还建议在现有的分布式计算环境中使用域分解技术实现并行算法。领域分解算法对于上述应用领域非常重要。

The use of parallel computers for solving partial differential equations is important in areas such as fluid dynamics, transonic flow, reservoir simulation, weather prediction and structural analysis, where many of the problems of interest can not be solved without the use of supercomputers. One technique for applying parallel computers to the solution of these problems is known as domain decomposition where the domain of interest is subdivided into several smaller subdomains and the task of solving the partial differential equation on each subdomain problem is assigned to a different processor. The global solution is then pieced together from the solutions computed on the individual subdomains. Much of the current work in the application of domain decomposition techniques has been in the area of elliptic partial differential equations with very little attention being given to hyperbolic equations. The investigator proposes to use the methods of domain decomposition for the solution of linear hyperbolic equations. The idea of using overlapping domains is introduced in the context of linear hyperbolic equations to develop a domain decomposition algorithm which is shown to be well suited for parallel processors. The questions of communication costs and load balancing is presented. The investigator also propose to implement parallel algorithms usong domain decomposition techniques in an existing distributed computing environment. Domain decomposition algorithm are important for the areas of applications mentioned above.