Vector and Parallel Algorithms for Singular Elliptic Partial Differential Equations

椭圆型偏微分方程是分析物理系统行为的重要工具。它们出现在研究扭转、不可压缩流体流动、电场和磁场引起的温度分布、应力和变形，以及许多其他科学和工程领域。通常，它们的解可以通过电子数字计算机获得，因此应用数学家、计算机科学家和工程师都对它们感兴趣。拟议研究的第一阶段将制定、实施并评估一种设计为对具有已知形式的奇性的椭圆边值问题最有效的有限差分方法。评估过程将考虑该方法在一系列问题上的准确性，以及确定其在一定网格大小范围内的计算效率。该项目的第二阶段将在Cray X-MP/24超级计算机上实现原始标量算法的矢量版和并行版。然后将研究增加的矢量长度和增加的多任务处理之间的性能权衡。

Elliptic partial differential equations are an important tool in the analysis of the behavior of physical systems. They arise in the study of temperature distribution, stress and deformation due to torsion, incompressible fluid flow, electric and magnetic fields, and in many other fields of science and engineering. Very often their solutions may be obtained with the aid of an electronic digital computer, and so they are of interest to applied mathematicians, computer scientists, and engineers alike. The first phase of the proposed research would formulate, implement, and then evaluate a finite difference method designed to be most effective on elliptic boundary-value problems with singularities of known form. The evaluation process would consider the method's accuracy over a population of problems, as well as determine its computational efficiency over a range of grid sizes. The second phase of the project would implement vector and parallel versions of the original scalar algorithm on a Cray X-MP/24 supercomputer. The performance tradeoffs between increased vector length and increased multitasking would then be investigated.