A parallel algorithmic framework for flexible time discretization adaptive Cartesian grids

灵活时间离散自适应笛卡尔网格并行算法框架

• 批准号: 1419108
• 负责人: Donna Calhoun
• 金额: $ 19.5万
• 依托单位: Boise State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1419108&HistoricalAwards=false
• 关键词: parallel; algorithmic; framework; flexible; time

Accurately predicting the weather, understanding global climate change, designing novel materials, developing means of exploiting energy resources, and modeling the effects of natural hazards increasingly rely on our ability to efficiently solve mathematical equations on large scale computing platforms. To exploit the emerging computing power now available on multi-core desktop machines as well as at local and national supercomputing centers, numerical algorithms originally designed to run on a single computing processor (e.g. a CPU) must often be redesigned to operate efficiently (or "scale") in a supercomputing environment with thousands of processors. This project focuses on redesigning a particular class of numerical methods that dynamically allocate computing resources to spatial regions of a computational domain where a simulation is most demanding. For example, such methods would place many more grid points (e.g. pixels) at a burning flame front, but leave the empty space in an industrial burner only coarsely resolved. Or, to accurately track a thin filament of volcanic ash, an adaptive method will update high resolution regions of the simulation domain to follow the ash plume as it meanders through the atmosphere, but will not waste computational resources in areas of the globe where no ash has arrived. Many such adaptive methods now show modest scalability in a multi-processor environment, but we propose a new software paradigm which will allow these "adaptive refinement mesh" methods to scale efficiently to ever larger numbers of computing processors as well as enable domain scientists to more easily incorporate complex numerical algorithms into a high performance software frameworks. Successful achievement of project goals will enable researchers to take advantage of the national investment in supercomputing centers and to make progress towards providing solutions to grand challenge problems. As a particular demonstration of our adaptive mesh paradigm, we will produce high resolution simulations of volcanic ash transport in the atmosphere. Such simulations are critical for predicting aviation hazards associated with volcanic eruptions.Single step, single stage multi-rate schemes are routinely used for solving partial differential equations on adaptively refined meshes. However, such methods are usually limited to second order accuracy or may suffer from operator splitting errors. Higher order temporal discretizations involving multiple stages or complex coupling strategies are considerably more difficult to incorporate into existing adaptive mesh software frameworks. The PI proposes a highly scalable algorithmic framework that simplifies the task of implementing sophisticated time stepping strategies into adaptive Cartesian mesh methods. The PI anticipates providing functionality that allows the user to describe their temporal strategy in a natural, method-of-lines setting. This will require designing an efficient, scalable data pipeline that provides a vectorized view of spatial data distributed across adaptively refined meshes and processors. Emphasis will be focused on explicit multi-stage Runge-Kutta methods for hyperbolic and parabolic conservation laws. Targeted applications of this work include the application of the theory of multi-rate methods for ODEs to the method-of-lines setting, the implementation of multi-rate, explicit Runge-Kutta-Chebyshev methods for reaction diffusion equations in an adaptive framework, and a demonstration of the effectiveness of the proposed framework on modeling dispersion of airborne volcanic ash in the atmosphere. The work will be done using the ForestClaw software platform, developed by the PI and her collaborator C. Burstedde (Univ. of Bonn, Germany).

准确预测天气、了解全球气候变化、设计新材料、开发能源开发手段以及模拟自然灾害的影响，越来越依赖于我们在大规模计算平台上有效求解数学方程的能力。为了利用现在在多核台式机上以及在本地和国家超级计算中心处可用的新兴计算能力，最初被设计为在单个计算处理器（例如，CPU）上运行的数值算法必须经常被重新设计以在具有数千个处理器的超级计算环境中有效地操作（或“扩展”）。这个项目的重点是重新设计一类特殊的数值方法，动态分配计算资源的空间区域的计算域的模拟是最苛刻的。 例如，这样的方法将在燃烧的火焰前缘处放置更多的网格点（例如，像素），但是在工业燃烧器中留下仅粗略解析的空白空间。 或者，为了精确地跟踪细丝状的火山灰，自适应方法将更新模拟域的高分辨率区域，以在火山灰羽流蜿蜒穿过大气时跟随火山灰羽流，但不会在没有火山灰到达的地球仪区域浪费计算资源。许多这样的自适应方法现在显示出适度的可扩展性在多处理器环境中，但我们提出了一个新的软件范例，这将允许这些“自适应细化网格”的方法，以有效地扩展到越来越多的计算处理器，以及使域科学家更容易地将复杂的数值算法到一个高性能的软件框架。项目目标的成功实现将使研究人员能够利用国家对超级计算中心的投资，并在为重大挑战问题提供解决方案方面取得进展。作为我们的自适应网格范例的一个特别演示，我们将产生高分辨率的火山灰在大气中的传输模拟。这种模拟对于预测与火山爆发有关的航空灾害是至关重要的。单步、单阶段多速率格式通常用于在自适应细化网格上求解偏微分方程。然而，这样的方法通常限于二阶精度或可能遭受算子分裂误差。涉及多个阶段或复杂耦合策略的高阶时间离散化相当难以纳入现有的自适应网格软件框架。PI提出了一个高度可扩展的算法框架，简化了自适应笛卡尔网格方法实现复杂的时间步进策略的任务。PI预期提供的功能，允许用户描述他们的时间策略，在一个自然的，方法线设置。这将需要设计一个有效的，可扩展的数据管道，提供分布在自适应细化网格和处理器的空间数据的矢量化视图。重点将集中在双曲型和抛物型守恒律的显式多级龙格-库塔方法。这项工作的有针对性的应用包括多速率方法的常微分方程的线设置的方法的理论的应用，多速率，显式龙格-库塔-切比雪夫方法的反应扩散方程的自适应框架中的实施，并演示了所提出的框架对空气中的火山灰在大气中的扩散建模的有效性。这项工作将使用由PI和她的合作者C开发的ForestClaw软件平台完成。Burstedde（波恩大学，德国）。