Innovative Time Integrators for Stiff and Highly Oscillatory Systems

科学和工程的进步越来越需要快速和准确的计算方法来模拟复杂的多物理过程及其在广泛的时间尺度或高频范围内的相互作用。数值天气预报和气候模拟是这类应用的著名例子，它们依赖于原始方程的计算解，这些原始方程用于预测大气、海洋、陆地表面、冰等的行为以及它们之间的复杂相互作用。这种多物理问题的数值解仍然是一项具有挑战性的任务，因为系统中存在多个时间尺度，其中不同的过程需要不同的时间来完成。因此，开发能够提供快速可靠解的先进数值方法对于许多依赖于复杂系统的大规模模拟的应用是至关重要的。该项目的总体目标是开发适用于刚性和高振荡系统的新的时间积分方法，并展示其在数值天气预报、海洋模拟和分子动力学模拟等应用中的性能。此外，该项目旨在培养一名博士生，并为密西西比州立大学的本科生和研究生提供数学方面的机会。预期至少对一名研究生进行关于拟议工作主题的培训。该项目有四个主要目标。首先，研究人员将为刚性系统导出新的混合指数积分器和预条件有理指数积分器并实现它们。其次，研究人员将为加性分区系统开发严格精度的嵌入式多速率指数方法。第三，研究人员将开发严格精确的指数Nyström方法。第四，研究人员将研究新开发的方法在数值天气预报、海洋建模和分子动力学模拟中的应用性能。研究人员将利用他以前在构建、分析和实施指数和多速率时间积分方法方面的专业知识来实现这些目标。与数值分析师、气象学家和计算机科学家的持续合作也将有助于该项目的成功。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。

Advancements in science and engineering increasingly require fast and accurate computational methods for simulating complex multi-physical processes and their interactions occurring at a wide range of temporal scales or high frequencies. Numerical weather prediction and climate modeling are notable examples of such applications that rely on the computational solution of primitive equations, which are used to predict the behavior of the atmosphere, oceans, land surface, ice, etc., as well as the complex interactions among them. Numerical solutions of such multiphysics problems remain a challenging task due to the presence of multiple time scales in the system where different processes take different amounts of time to complete. As such, developing advanced numerical methods capable of offering fast and reliable solutions is crucial for many applications that rely on large-scale simulations of complex systems. The overall goal of this project is to develop novel time integration methods for stiff and highly oscillatory systems and demonstrate their performance on applications such as numerical weather prediction, ocean modeling, and molecular dynamics simulations. Additionally, the project aims to train one doctoral student and offer opportunities to undergraduate and graduate students in mathematics at Mississippi State University. Training of at least one graduate student on the topics of the proposed work is expected. The project has four primary aims. First, the investigator will derive novel mixed exponential integrators and preconditioned rational exponential integrators for stiff systems and implement them. Second, the investigator will develop stiffly-accurate embedded multirate exponential methods for additively partitioned systems. Third, the investigator will develop stiffly-accurate exponential Nyström methods. Fourth, the investigator will investigate the performance of the newly developed methods on applications in numerical weather prediction, ocean modeling, and molecular dynamics simulations. The investigator will build off of his previous expertise in constructing, analyzing, and implementing exponential and multirate time integration methods to achieve these aims. Ongoing collaborations with numerical analysts, meteorologists, and computer scientists will also contribute to the success of the project.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.