CAREER: Efficient and Accurate Local Time-Stepping Algorithms for Multiscale Multiphysics Systems

多尺度多物理场过程的数学建模和数值模拟非常重要，但由于各种过程发生在不同的尺度上并且相互耦合，因此具有很高的挑战性。在处理复杂的大规模系统（例如，在海洋和海岸建模中产生的那些系统）时，这些问题更为关键。本项目旨在提高多尺度多物理场系统的局部时间步进算法的效率和保真度，并将其应用于大尺度地球物理流的多分辨率模拟。所开发的算法将有效地捕获空间和时间上的大范围尺度，从而在很长一段时间内对这些系统进行准确和稳健的模拟。该研究计划与该项目的教育活动紧密结合，其中包括(i)在奥本大学夏季科学研究所开发计算数学课程模块，这是一个针对高中生的教育丰富计划，为年轻学生提供早期应用数学的机会，并激励他们追求科学，技术，工程和数学（STEM）的职业生涯；（ii）为本科生和研究生提供跨学科的应用数学教育和研究培训，包括妇女和代表性不足的少数民族。从技术上讲，首席研究员将开发基于非重叠域分解的准确有效的混合局部时间步进算法：一方面，具有局部时间步长的显式方案用于小时间尺度上的过程建模，而不会受到全局CFL条件所规定的时间步长的严格限制。另一方面，局部指数时间积分器被用来实现在缓慢速度下发生的过程的大时间步长，并通过局部并行执行这些计算来加速矩阵指数及其乘积的计算。主要有三个研究目标：(i)发展和分析刚性非线性方程的非重叠局部指数时差方法；（ii）研究各种异构问题的混合局部时间步进算法；(3)这些算法在模拟海洋/大气环流的三维原始方程中的应用。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。

Mathematical modeling and numerical simulations of multiscale multiphysics processes are of great importance, yet highly challenging as various processes occur at different scales and are coupled together. The issues are more crucial when dealing with complex large-scale systems (for example, those arising in ocean and coastal modeling). The goal of this project is to advance the efficiency and fidelity of local time-stepping algorithms for multiscale multiphysics systems with application to multi-resolution simulations of large-scale geophysical flows. The developed algorithms will efficiently capture the wide range of scales in both space and time to produce accurate and robust simulations of these systems over a long period of time. The research plan is closely integrated with the educational activities of the project which include (i) developing curricular modules in computational mathematics at the Auburn University Summer Science Institute, an educational enrichment program for high school students, to provide young students early exposure to applied mathematics and inspire them to pursue a career in Science, Technology, Engineering and Mathematics (STEM); and (ii) providing interdisciplinary applied mathematics education and research training for both undergraduate and graduate students, including women and underrepresented minorities.Technically, the Principal Investigator will develop accurate and effective hybrid local time-stepping algorithms based on nonoverlapping domain decomposition: on the one hand, explicit schemes with local time steps are used to model processes at small time scales without suffering a severe restriction on the time step size dictated by the global CFL condition. On the other hand, localized exponential time integrators are employed to enable large time step sizes for processes occurring at slow speeds, and to accelerate the computation of matrix exponentials and their products by performing these calculations locally and in parallel. Three main research objectives will be pursued: (i) development and analysis of nonoverlapping localized exponential time differencing methods for stiff nonlinear equations; (ii) study of hybrid local time-stepping algorithms for various heterogeneous problems; and (iii) application of these algorithms to the three-dimensional primitive equations for modeling ocean/atmosphere circulations.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.