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: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=548513
• 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.

从数学上讲，工程师，经济学家和科学家感兴趣的许多模型都是作为部分微分方程（PDE）写的。 除了某些理想化的情况外，无法通过分析解决的PDE。相反，我们依靠数值近似值。 我对在两个或三个空间维度的时间依赖性PDE的溶液的有效实施和自适应算法的研究特别感兴趣，其解决方案表现出较大的溶液变化，奇异性形成或移动前部。我们研究试图通过在解决方案具有有趣但难以跟踪行为的区域中集中计算工作（在时空和时间上）来有效地获得解决方案的方法。该策略通过使用移动空间网格来跟踪解决方案的有趣特征，从而在时间前进。 此外，还有一个机会和动力来研究旨在利用不断发展的计算硬件的算法 - 容易获得的商品集群，具有数百或数千个核心，混合CPU-GPU系统，甚至具有4-24个核心的台式机。我们通过将大问题分为在单个内核上计算的小块并重新组合以使用域分解（DD）算法来解决原始问题的解决方案，从而将依赖性PDE的解决方案映射到多核环境。 这种策略将用于自适应网格的产生以及物理PDE的解决方案。 可以通过计算同时预测和校正来添加时间的小规模并行性。最终，我们将为适用于现有和新兴HPC硬件的时间依赖性PDE的并行自适应解决方案提供一个新的基于理论上的模块化平台。该研究计划为移动网格方法提供了影响的途径，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HQP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。