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Collaborative Research: Euler-Based Time-Stepping with Optimal Stability and Accuracy for Partial Differential Equations

协作研究：具有最佳稳定性和精度的偏微分方程基于欧拉的时间步进

基本信息

批准号：
2012271
负责人：
Benjamin Seibold
金额：
$ 10万
依托单位：
Temple University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-15 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012271&HistoricalAwards=false
关键词：
Collaborative Research Euler Based Time

项目摘要

Principled simulations of many real-world problems (such as fluid flow, geophysical phenomena, and quantum mechanics) require an evolution in time with high accuracy, yet in a structurally simple and robust fashion. This project develops novel time integration methods for complex multi-physics problems, while not incurring fundamental problems that reduce the accuracy or stability of many existing methods. The developed methods are founded in new mathematical theories, and are used to devise more accurate and robust simulations of shallow water flows with dispersive effects, which are important in the understanding of tsunamis, storm surge, and coastal flooding. This project will support one graduate student for two years at NJIT and one graduate student per year at the second institution, Temple.This project develops methods for the time integration of differential equations that are implemented as sequences of generalized Euler steps, including: multistage diagonally implicit Runge-Kutta (DIRK) and multistep implicit-explicit (IMEX) methods. Such methods are significant as they reduce the implementation burden on a practitioner to the solution of a fully- or semi-implicit Euler step for their initial-boundary-value problem. The key research contributions are: (A) a full algebraic theory of weak stage order, and its use to design optimized high-order DIRK methods devoid of order reduction; (B) a stability theory for IMEX methods applied to differential algebraic equations, and the co-design of IMEX splittings and scheme coefficients to overcome stability limitations prevalent in existing methods. Applications include new efficient time-stepping for the dispersive shallow water equations and related differential algebraic equations. The collaborative mentoring of graduate students at two campuses is an important component of this 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.

许多现实世界问题（如流体流动，地球物理现象和量子力学）的原理模拟需要高精度的时间演化，但在结构上简单和强大的方式。该项目为复杂的多物理问题开发了新的时间积分方法，同时不会导致降低许多现有方法的准确性或稳定性的基本问题。所开发的方法建立在新的数学理论，并用于设计更准确和强大的模拟浅水流动与分散的影响，这是重要的海啸，风暴潮和沿海洪水的理解。该项目将资助一名研究生在NJIT学习两年，每年资助一名研究生在第二所大学Temple学习。该项目开发了微分方程的时间积分方法，这些方法被实现为广义欧拉步骤序列，包括：多级对角隐式Runge-Kutta（DIRK）和多步隐式显式（IMEX）方法。这样的方法是重要的，因为它们减少了执行负担的从业者的解决方案的完全或半隐式欧拉步骤，他们的初边值问题。主要的研究贡献是：（A）一个完整的代数理论的弱阶，并使用它来设计优化的高阶DIRK方法没有降阶;（B）的IMEX方法应用于微分代数方程的稳定性理论，IMEX分裂和计划系数的协同设计，以克服现有方法中普遍存在的稳定性限制。应用包括新的有效的时间步长的色散浅水方程和相关的微分代数方程。两个校区的研究生的合作指导是该项目的重要组成部分。该奖项反映了NSF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。