XPS: FULL: DSD: Asynchronous PDE Algorithms for Turbulent Flows at Exascale

未来的亿级计算系统将可用于研究重要的计算密集型应用，如多物理、多尺度的自然现象和工程系统，通常由偏微分方程组(PDE)精确建模。一个主要的例子是高雷诺数的湍流，通常存在于自然和工程系统中，它包括非常广泛的空间和时间尺度，因此已成为科学计算中的一项重大挑战。在有效利用亿级系统之前，存在许多必须克服的挑战。这包括处理单元之间的通信以及全局同步，这两者都可能成为在模拟中使用数百万个处理单元时的主要瓶颈。在这个项目中，PI为偏微分方程组开发了新的艾级数值格式，特别是那些描述湍流的偏微分方程组，它利用了从数学到软件水平的异步。它们基于广泛使用的有限差分、紧致微分和频谱格式。异步提供了更好的性能，但也在解中引入了误差。新方案将能够以定量和可预测的方式权衡精确度和性能。方法包括：(I)基于数值分析和动力系统的稳定性和精确度的严格数学研究，这也将为开发新方案和量化其不确定性提供框架；(Ii)开发可扩展库中的特定元素用于并行计算，以便能够在当前和未来的机器上进行可移植实施；以及(Iii)基于物理的对不现实流动的数值扰动的建模。该项目中的工具、技术和模拟数据将通过研究生指导、本科生研究和作为高性能计算课程的材料被整合到PIS的教育工作中，被PI捕获的流体动力学和动力系统。

Future exascale computing systems will be available to study important,compute-intensive applications such as multi-physics multi-scale naturalphenomena and engineering systems typically modeled accurately by partialdifferential equations (PDEs).  A prime example is turbulence at high Reynoldsnumbers, typically found in natural and engineering systems, which comprise anextremely wide range of spatial and temporal scales and has thus became a GrandChallenge in scientific computing.Many challenges exists that must be overcome before exascale systemscan be utilized effectively. These include communication betweenprocessing elements as well as global synchronizations both of which willlikely be a main bottleneck when millions of billions of processing elementsare utilized in a simulation.In this project, the PIs develop novel exascale numerical schemes for PDEs,especially those describing turbulent flows, that exploit asynchrony from the mathematical tothe software level. These are based on widely used finite differences, compact differentiation and spectral schemes.Asynchrony offers better performance but also introduces errors in the solution. The new schemes willbe able to trade-off accuracy and performance in a quantitative and predictable manner.The approach includes (i) rigorous mathematical studies of stability andaccuracy based on numerical analysis and dynamical systems, which will alsoprovide a framework for the development of new schemes and quantify itsuncertainty, (ii) development of specific elements in a scalable library for parallelcomputing to enable portable implementations on current and future machines,and (iii) physics based modeling of numerical perturbations inrealistic flows.The tools, techniques and simulation data in this projectwill be integrated in the PIs' educational efforts through graduate mentoring, undergraduateresearch and as material for courses in high-performance computing, fluid dynamics and dynamical systemstaught by the PIs.