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

许多现实世界问题（如流体流动、地球物理现象和量子力学）的原则模拟需要高精度的时间演化，但结构简单且健壮。该项目为复杂的多物理场问题开发了新颖的时间积分方法，同时不会产生降低许多现有方法精度或稳定性的根本问题。所开发的方法建立在新的数学理论基础上，并用于设计更准确、更可靠的具有分散效应的浅水流动模拟，这对理解海啸、风暴潮和沿海洪水非常重要。该项目将在新泽西理工大学支持一名研究生两年，在第二所大学坦普尔大学每年支持一名研究生。该项目开发了微分方程的时间积分方法，这些方法实现为广义欧拉步骤序列，包括：多阶段对角隐式龙格-库塔（DIRK）和多步骤隐式-显式（IMEX）方法。这些方法的意义在于，它们减轻了实践者解决其初边值问题的全隐式或半隐式欧拉步骤的实现负担。主要研究成果有：(A)建立了弱阶段阶的完整代数理论，并将其用于设计无阶约化的高阶DIRK优化方法；(B)应用于微分代数方程的IMEX方法的稳定性理论，以及IMEX分裂和方案系数的协同设计，以克服现有方法中普遍存在的稳定性限制。应用包括对色散浅水方程和相关微分代数方程的新的有效时间步进。两个校区的研究生合作指导是该项目的重要组成部分。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。

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.