Collaborative Research: Parallel Space-Time Solvers for Systems of Partial Differential Equations

合作研究：偏微分方程组的并行时空求解器

• 批准号: 2110917
• 负责人: Jacob Schroder
• 金额: $ 15.25万
• 依托单位: University of New Mexico
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2110917&HistoricalAwards=false
• 关键词: Collaborative; Research; Parallel; Space; Time

Computer simulations and the mathematical methods supporting these are central to the modern study of engineering, biology, chemistry, physics, and other fields. Many simulations are computationally costly and require the large resources of modern supercomputers. New mathematical methods are urgently needed to efficiently utilize next generation supercomputers with millions to billions of processors. This project will develop new parallel-in-time algebraic multigrid methods for complex physical systems specifically designed for next generation computers. These new methods will add a new dimension of parallel scalability (time) and promise dramatically faster simulations in many important application areas, such as the gas and fluid dynamics problems considered (e.g., with relevance to wind turbines and viscoelastic flow). Graduate students will be involved and trained, and open source code will be developed.This project will develop fast, parallel, and flexible space-time solvers for systems of partial differential equations (PDEs). The project will focus on algebraic multigrid (AMG) within block preconditioning traditionally appropriate for large adaptively refined spatial systems. These techniques will be extended to general space-time systems with a flexible approach that allows for adaptive space-time refinement. This adaptivity helps to accurately resolve lower dimensional features such as shocks at a fraction of the cost and storage of uniform refinement. Furthermore, the project will produce new practical AMG theory for non-SPD (symmetric positive definite) problems as well as solvers for adaptively refined space-time discretizations for a variety of parabolic and hyperbolic PDEs including the Euler and Navier-Stokes equations and Cahn-Hilliard system. The project will design, analyze, and tune parallel AMG solvers that are robust, efficient, and fast over a wide range of PDEs and parameters and will contribute to the widely used packages MFEM and hypre. The solvers will be developed and tested for applications in wind turbines, as well the high Weissenberg number problem in viscoelastic flows.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

计算机模拟和支持这些模拟的数学方法是现代工程、生物学、化学、物理学和其他领域研究的核心。许多模拟的计算成本很高，并且需要现代超级计算机的大量资源。迫切需要新的数学方法来有效地利用具有数百万至数十亿处理器的下一代超级计算机。 该项目将为专为下一代计算机设计的复杂物理系统开发新的时间并行代数多重网格方法。 这些新方法将增加并行可扩展性（时间）的新维度，并有望在许多重要应用领域显着加快模拟速度，例如所考虑的气体和流体动力学问题（例如，与风力涡轮机和粘弹性流相关）。研究生将参与并接受培训，并将开发开源代码。该项目将为偏微分方程（PDE）系统开发快速、并行且灵活的时空求解器。该项目将重点关注传统上适用于大型自适应细化空间系统的块预处理内的代数多重网格（AMG）。这些技术将通过灵活的方法扩展到通用时空系统，从而实现自适应时空细化。这种适应性有助于以均匀细化的成本和存储的一小部分准确地解析较低维的特征，例如冲击。此外，该项目还将针对非 SPD（对称正定）问题产生新的实用 AMG 理论，以及针对各种抛物线和双曲偏微分方程（包括欧拉和纳维-斯托克斯方程以及 Cahn）自适应细化时空离散化的求解器。希利亚德系统。该项目将设计、分析和调整并行 AMG 求解器，这些求解器在各种偏微分方程和参数上稳健、高效且快速，并将为广泛使用的 MFEM 和 hypre 软件包做出贡献。这些求解器将针对风力涡轮机的应用以及粘弹性流中的高魏森伯格数问题进行开发和测试。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。