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

计算机模拟和支持这些的数学方法是工程、生物、化学、物理和其他领域的现代研究的中心。许多模拟在计算上是昂贵的，并且需要现代超级计算机的大量资源。为了有效地利用拥有数百万到数十亿处理器的下一代超级计算机，迫切需要新的数学方法。这个项目将开发新的时间并行代数多重网格方法，用于专门为下一代计算机设计的复杂物理系统。这些新方法将增加并行可伸缩性(时间)的新维度，并承诺在许多重要应用领域进行显著更快的模拟，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动相关)。该项目将开发快速、并行和灵活的偏微分方程组(PDE)时空解算器。该项目将专注于块预适应中的代数多重网格(AMG)，传统上适用于大型自适应细化空间系统。这些技术将扩展到一般的时空系统，具有灵活的方法，允许自适应的时空精化。这种自适应能力有助于准确地解决低维特征，如冲击，而成本和存储统一细化的成本只有很小的一部分。此外，该项目将产生用于非SPD(对称正定)问题的新的实用AMG理论，以及用于各种抛物型和双曲型偏微分方程组(包括Euler和Navier-Stokes方程以及Cahn-Hilliard系统)的自适应细化时空离散的求解器。该项目将设计、分析和调整并行AMG解算器，这些解算器在广泛的偏微分方程和参数范围内是健壮、高效和快速的，并将有助于广泛使用的软件包MfeM和HyPRE。这些解算器将被开发并测试在风力涡轮机中的应用，以及粘弹性流动中的高魏森伯格数问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.