A parallel algorithmic framework for flexible time discretization adaptive Cartesian grids

准确预测天气、了解全球气候变化、设计新材料、开发开发能源资源的方法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高效求解数学方程的能力。为了利用目前在多核台式机以及地方和国家超级计算中心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高效地运行(或“扩展”)。这个项目专注于重新设计一类特定的数值方法，这些方法将计算资源动态分配到计算领域中对模拟要求最高的空间区域。例如，这种方法会在燃烧的火焰前沿放置更多的网格点(例如像素)，但只会粗略地解决工业燃烧器中的空白空间。或者，为了准确跟踪细丝火山灰，一种自适应方法将更新模拟域的高分辨率区域，以跟踪火山灰羽流在大气中盘旋时的轨迹，但不会在地球上没有火山灰到达的地区浪费计算资源。许多这样的自适应方法在多处理器环境中表现出适度的可扩展性，但我们提出了一种新的软件范例，它将允许这些“自适应精化网格”方法有效地扩展到越来越多的计算处理器，并使领域科学家能够更容易地将复杂的数值算法整合到高性能的软件框架中。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殊演示，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喷发有关的航空危险是至关重要的。单步、单级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组。然而，这种方法通常局限于二阶精度，或者可能受到算子分裂误差的影响。涉及多阶段或复杂耦合策略的高阶时间离散化很难结合到现有的自适应网格软件框架中。PI提出了一个高度可扩展的算法框架，简化了将复杂的时间推进策略实现为自适应笛卡尔网格方法的任务。PI期望提供允许用户在自然的线条方法设置中描述其时间策略的功能。这将需要设计一个高效的、可伸缩的数据管道，为分布在自适应改进的网格和处理器上的空间数据提供矢量化的视图。重点将集中在双曲型和抛物型守恒律的显式多阶段Runge-Kutta方法。这项工作的目标应用包括将常微分方程组的多速率方法理论应用于直线方法的设置，在自适应框架中实现反应扩散方程的多速率显式Runge-Kutta-Chebyshev方法，以及演示所提出的框架在模拟大气中气载火山灰扩散方面的有效性。这项工作将使用由PI及其合作者C.Burstedde(Univ.)开发的ForestClaw软件平台完成。来自德国波恩)。

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).