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现代工程学、生物学、化学、物理学和其他领域研究的核心。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。迫切需要新的数学方法来有效地利用具有数百万到数十亿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，并承诺在许多重要应用领域中显著更快的模拟，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，并行和灵活的时空求解器。该项目将集中在代数多重网格（AMG）块预处理传统上适用于大型自适应细化空间系统。这些技术将被扩展到一般的时空系统的灵活的方法，允许自适应时空细化。这种自适应性有助于准确地解决低维的功能，如冲击的成本和存储的均匀细化的一小部分。此外，该项目将为非SPD（对称正定）问题产生新的实用AMG理论，并为各种抛物和双曲PDE（包括Euler和Navier-Stokes方程和Cahn-Hilliard系统）的自适应精化时空离散化提供解决方案。该项目将设计，分析和调整并行AMG求解器，这些求解器在广泛的PDE和参数范围内具有鲁棒性，高效性和快速性，并将为广泛使用的软件包MFEM和hypre做出贡献。该解决方案将被开发和测试用于风力涡轮机的应用，以及粘弹性流中的高Weissenberg数问题。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.