New Advanced Time Integration Methods for Multiphysics Systems and Applications

科学和工程中许多重要的系统都涉及多个物理过程。这些组件之间复杂的相互作用可能导致在大范围的时间尺度上的动态，这一特征对计算模拟提出了重大挑战。突出的例子来自大气现象的巨大时间尺度差异，这是数值天气预报（NWP）和气候模拟的核心问题。在耦合振荡器（如纤维、流体或柔性固体）的动力学计算建模中也出现了类似的挑战，在动画和医学成像中有许多应用。该项目的目标是为复杂耦合模式开发新的快速准确的时间积分方法，并将其应用于NWP和气候模拟以及其他实例。在教育方面，该项目将直接涉及和培养一名博士生，并为其他数学本科生和研究生提供研究机会。本项目考虑一般耦合多物理场系统。目标是为多物理场模型开发新的先进指数Rosenbrock/Runge-Kutta和多速率时间积分方法。该项目有三个研究目标。首先，PI计划为刚性多物理场系统推导新的并行和自适应指数方法。有了这一点，PI的目标是推导出新的时间步进方法，优化结构和局部误差控制以提高效率。其次，PI计划开发一个完整的理论，用于构建两类新的多速率方法，用于划分多物理场系统。第三，在现有代码和与数值分析师、气象学家和计算机科学家的持续合作的基础上，PI将调查新开发的方法在前面提到的应用程序中的性能。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。

Many important systems in science and engineering involve multiple physical processes. Complex interactions between these components can result in dynamics over a wide range of time scales, and this feature poses significant challenges for computational simulations. Prominent examples come from the vastly differing time scales in atmospheric phenomena, a central issue of numerical weather prediction (NWP) and climate modeling. Similar challenges arise in computational modeling of dynamics of coupled oscillators such as fibers, fluids, or flexible solids, with many applications in animation and medical imaging. The goal of this project is to develop new fast and accurate time integration methods for complex coupled models and to apply them to NWP and climate simulations as well as to other examples. On the educational side, the project will directly involve and train one doctoral student, as well as offer research opportunities to other undergraduate and graduate students in mathematics.This project considers general coupled multiphysics systems. The goal is to develop new advanced exponential Rosenbrock/Runge-Kutta and multi-rate time integration methods for multiphysics models. The project has three research objectives. First, the PI plans to derive new parallel and adaptive exponential methods for stiff multiphysics systems. With this in place, the PI aims to derive new time-stepping methods that have optimized structures and local error control for efficiency. Second, the PI plans to develop a complete theory for constructing two new classes of multirate methods for partitioned multiphysics systems. Third, building on existing codes and ongoing collaborations with numerical analysts, meteorologists, and computer scientists, the PI will investigate the performance of newly-developed methods on applications mentioned previously.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.