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的数学方法是对工程，生物学，化学，物理和其他领域的现代研究的核心。许多模拟在计算上都是昂贵的，需要大量现代超级计算机的资源。迫切需要新的数学方法来有效利用具有数百万到数十亿个处理器的下一代超级计算机。 该项目将为专门为下一代计算机设计的复杂物理系统开发新的平行代数多机方法。 这些新方法将在许多重要的应用领域，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程相关），在许多重要的应用领域（例如，与气体和流体动力学问题相关），将增加并行可伸缩性（时间）的新维度。研究生将参与和培训，并将开发开源代码。该项目将开发快速，平行且灵活的时空求解器，用于偏微分方程（PDES）的系统。该项目将集中于传统上适用于大型适应性精制空间系统的块预处理中的代数多机（AMG）。这些技术将以灵活的方法扩展到一般的时空系统，从而可以进行自适应时空的改进。这种适应性有助于准确地解决较低的维度特征，例如均匀精炼成本和存储的一小部分冲击。此外，该项目将针对非SPD（对称的确定）问题以及用于适应性精制的时空离散器的新实用AMG理论，用于各种抛物线和双曲线PDE，包括Euler和Navier-Stokes方程以及Cahn-Hilliard System。该项目将在广泛的PDE和参数上设计，分析和调整平行的AMG求解器，这些求解器稳健，高效且快速，并将为广泛使用的包装MFEM和HYPRE做出贡献。将开发和测试求解器在风力涡轮机中的应用，以及粘弹性流中的高魏森堡数量问题。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估来评估的。