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、开发能源开发手段以及对自然灾害影响进行建模，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。为了利用多核台式计算机以及地方和国家超级计算中心的新兴计算能力，最初设计在单个计算处理器（例如CPU）上运行的数值算法必须经常重新设计，以便在具有数千个处理器的超级计算环境中有效地运行（或“扩展”）。该项目的重点是重新设计一类特定的数值方法，动态地将计算资源分配到最需要模拟的计算域的空间区域。例如，这样的方法会在燃烧的火焰前放置更多的网格点（例如像素），但在工业燃烧器中留下的空白空间只能粗略地解决。或者，为了精确地追踪火山灰细丝，一种自适应方法将更新模拟域的高分辨率区域，以跟踪在大气中蜿蜒移动的火山灰羽流，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现在在多处理器环境中显示出适度的可扩展性，但我们提出了一种新的软件范例，它将允许这些“自适应细化网格”方法有效地扩展到更大数量的计算处理器，并使领域科学家更容易将复杂的数值算法合并到高性能软件框架中。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。作为我们自适应网格范例的一个特殊演示，我们将产生大气中火山灰运输的高分辨率模拟。这种模拟对于预测与火山爆发有关的航空危害至关重要。单步、单阶段、多速率格式通常用于求解自适应精细网格上的偏微分方程。然而，这种方法通常限于二阶精度或可能遭受算子分裂误差。涉及多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，简化了将复杂的时间步进策略实现到自适应笛卡尔网格方法中的任务。PI期望提供的功能允许用户在自然的、行法设置中描述他们的时间策略。这需要设计一个高效的、可扩展的数据管道，提供空间数据的矢量化视图，分布在自适应细化的网格和处理器之间。重点将集中在双曲和抛物守恒律的显式多阶段龙格-库塔方法上。本工作的目标应用包括将ode的多速率方法理论应用于线法设置，在自适应框架中实现反应扩散方程的多速率、显式龙格-库塔-切比雪夫方法，以及证明所提出的框架在模拟大气中空气中火山灰扩散方面的有效性。这项工作将使用由PI和她的合作者C. Burstedde（德国波恩大学）开发的ForestClaw软件平台来完成。

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).