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核心。许多模拟计算成本很高，需要现代超级计算机的大量资源。为了有效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。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，并有望在许多重要的应用领域显著加快模拟速度，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动相关）。研究生将参与并接受培训，开放源代码将被开发。本项目将为偏微分方程（PDEs）系统开发快速、并行和灵活的时空求解器。该项目将侧重于块预处理中的代数多网格（AMG），传统上适用于大型自适应精细空间系统。这些技术将通过允许自适应时空细化的灵活方法扩展到一般时空系统。这种适应性有助于准确地解决较低维度的特征，如冲击，成本和均匀细化存储的一小部分。此外，该项目将为非spd（对称正定）问题提供新的实用AMG理论，并为各种抛物型和双曲型偏微分方程（包括Euler和Navier-Stokes方程以及Cahn-Hilliard系统）提供自适应改进的时空离散解。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偏微分方程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MFEM和hyperc软件包做出贡献。求解器将被开发和测试用于风力涡轮机的应用，以及粘弹性流动中的高Weissenberg数问题。该奖项反映了美国国家科学基金会的法定使命，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.