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

多尺度多物理过程的数学建模和数值模拟是非常重要的，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，海洋和沿海模拟中出现的系统)时，这些问题更为关键。本项目的目标是提高多尺度多物理系统的局部时间步长算法的效率和保真度，并将其应用于大尺度地球物理流动的多分辨率模拟。开发的算法将有效地捕捉空间和时间上的大范围尺度，以在很长一段时间内产生这些系统的准确和健壮的仿真。该研究计划与该项目的教育活动紧密结合在一起，包括(I)在奥本大学暑期科学学院开发计算数学课程模块，这是一项针对高中生的教育充实计划，为年轻学生提供早期接触应用数学的机会，并激励他们从事科学、技术、工程和数学(STEM)职业；以及(Ii)为本科生和研究生(包括女性和代表性不足的少数民族)提供跨学科的应用数学教育和研究培训。在技术上，首席调查员将开发基于非重叠区域分解的准确和有效的混合局部时间步长算法：一方面，使用具有局部时间步长的显式格式来模拟小时间尺度的过程，而不受全球CFL条件对时间步长的严格限制。另一方面，使用局部指数时间积分器来实现低速过程的大时间步长，并通过本地和并行地执行这些计算来加速矩阵指数及其乘积的计算。将追求三个主要研究目标：(I)刚性非线性方程的非重叠局部指数时差方法的发展和分析；(Ii)各种非均匀问题的混合局部时间推进算法的研究；以及(Iii)将这些算法应用于模拟海洋/大气环流的三维原始方程。该奖项反映了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，被认为是值得支持的。

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.