Parallel Space-Time Approaches for the Numerical Solution of Partial Differential Equations

偏微分方程数值解的并行时空方法

• 批准号: 311796-2013
• 负责人: Haynes, Ronald
• 金额: $ 2.19万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635131
• 关键词: Parallel; Space; Time; Approaches; Numerical

Mathematically, many models of interest to engineers, economists and scientists are written as partial differential equations (PDEs). Except for certain idealized situations, the PDEs which result are not possible to solve analytically. Instead we rely on numerical approximations. I am particularly interested in the study of efficient implementations and analysis of adaptive algorithms for the solution of time-dependent PDEs in two or three spatial dimensions whose solutions exhibit large solution variation, singularity formation or moving fronts. We study methods which attempt to obtain a solution efficiently by concentrating computational effort (in both space and time) in regions where the solution has interesting but difficult to track behavior. The strategy works by using moving spatial meshes to track interesting features of the solution forward in time. Moreover, there is an opportunity and motivation to study algorithms designed to take advantage of the continually evolving computing hardware - readily available commodity clusters with hundreds or thousands of cores, hybrid CPU-GPU systems and even desktop machines with 4-24 cores. We propose mapping the solution of time dependent PDEs to multi-core environments by dividing the large problem into small pieces computed on individual cores and recombined to give a solution of the original problem using domain decomposition (DD) algorithms. Such a strategy will be used for both the generation of the adaptive grids, as well as the solution of the physical PDE. Small scale parallelism in time may be added by computing simultaneous predictions and corrections. Ultimately, we will provide a new, theoretically based, modular platform for the parallel adaptive solution of time dependent PDEs suitable for existing and emerging HPC hardware. This research program provides a route to impact for moving mesh methods, providing a software tool for computational scientists for the solution of complex problems. Theoretically it will enhance our knowledge of the behaviour of DD algorithms for nonlinear problems. Finally, it will provide HQP with mathematical expertise and computational competency - transferable skills highly sought by employers.

在数学上，工程师、经济学家和科学家感兴趣的许多模型都被写成偏微分方程 (PDE)。除了某些理想化情况外，所产生的偏微分方程不可能通过分析求解。相反，我们依赖于数值近似。我对研究自适应算法的有效实现和分析特别感兴趣，该算法用于解决两个或三个空间维度中的时间相关偏微分方程，其解表现出较大的解变化、奇点形成或移动前沿。我们研究的方法试图通过将计算工作（在空间和时间上）集中在解决方案具有有趣但难以跟踪行为的区域来有效地获得解决方案。该策略的工作原理是使用移动的空间网格来及时跟踪解决方案的有趣特征。此外，还有机会和动机来研究旨在利用不断发展的计算硬件的算法——具有数百或数千个核心的现成商品集群、混合 CPU-GPU 系统，甚至具有 4-24 个核心的台式机。我们建议将时间相关 PDE 的解决方案映射到多核环境，方法是将大问题划分为在各个核心上计算的小块，并使用域分解 (DD) 算法重新组合以给出原始问题的解决方案。这种策略将用于自适应网格的生成以及物理偏微分方程的求解。可以通过计算同时的预测和校正来添加小规模的时间并行性。最终，我们将为适用于现有和新兴 HPC 硬件的时间相关 PDE 的并行自适应解决方案提供一个新的、基于理论的模块化平台。该研究计划为移动网格方法提供了一条影响途径，为计算科学家提供了解决复杂问题的软件工具。从理论上讲，它将增强我们对非线性问题 DD 算法行为的了解。最后，它将为总部提供数学专业知识和计算能力——雇主高度寻求的可转移技能。