New Advanced Time Integration Methods for Multiphysics Systems and Applications

科学和工程中的许多重要系统都涉及多个物理过程。这些组件之间的复杂交互可能会导致大范围的时间范围内的动力学，这一功能给计算模拟带来了巨大的挑战。突出的例子来自大气现象中差异巨大的时间尺度，这是数值天气预报(NWP)和气候建模的核心问题。类似的挑战出现在耦合振子的动力学计算建模中，如纤维、流体或柔性固体，在动画和医学成像中有许多应用。该项目的目标是开发用于复杂耦合模式的新的快速和准确的时间积分方法，并将其应用于数值预报和气候模拟以及其他例子。在教育方面，该项目将直接涉及和培养一名博士生，并为其他本科生和研究生提供研究机会。该项目考虑的是一般耦合多物理系统。其目标是开发用于多物理模型的新的先进的指数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计划为刚性多物理系统推导出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，这些方法具有优化的结构和局部误差控制，以提高效率。其次，PI计划发展一套完整的理论来构建两类新的用于分区多物理系统的多速率方法。第三，在现有代码的基础上，以及与数值分析师、气象学家和计算机科学家的持续合作，PI将调查新开发的方法在前面提到的应用程序中的表现。这一奖项反映了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。

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.