Collaborative Research: Accuracy-Preserving Robust Time-Stepping Methods for Fluid Problems

对基于物理或工程原理的现实模型的模拟通常需要求解大型的时间演化方程。该项目旨在开发新的数学理论和方法，以提高数值方法的准确性和效率，使这些问题在时间上向前推进，同时保持已有方法的一般结构。该研究包括适合于求解波动方程的方法，如声学或电磁学中出现的方法，以及带约束的方程，如流体流动模拟中出现的方程。一个关键的应用将是增强的浅水流体运动模拟，它可以描述风暴潮、海啸和通过极端天气(例如飓风)产生的波浪等现象。这个项目将支持两个研究生，他们将由两个机构的教师共同指导，还包括一个本科生研究部分。这个项目旨在建立微分方程时间积分的新方向，包括开发避免降阶的龙格-库塔方法和微分代数方程的多步方法。主要的研究贡献将是(A)为满足弱阶段序的显式和隐式-显式Runge-Kutta方法发展新的构造方法和代数理论；(B)证明刚性问题的时间步进障碍定理；以及(C)将所开发的方法发展到社区软件中。这些概念将被用来在三个特定的流体流动应用中提供新的方法：(I)色散浅水方程；(Ii)不可压缩的Navier-Stokes方程；以及(Iii)以对流为主的问题，包括双曲守恒定律。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，被认为值得支持。

The simulation of principled physics- or engineering-based models of reality often requires the solution of large time-evolution equations. This project aims to develop new mathematical theory and methodologies that improve the accuracy and efficiency of numerical approaches to time-advance such problems, while retaining the general structure of established methods. The research includes methods appropriate for solving wave equations, such as those arising in acoustics, or electromagnetic, and equations with constraints, such as those arising in the simulation of fluid flows. A key application will be the enhanced simulation of shallow water fluid motion, which can describe phenomena such as storm surge, tsunamis, and wave generation via extreme weather (e.g., hurricanes). This project will support two graduate students who will be co-mentored by faculty at two institutions, and also include an undergraduate research component.This project aims to establish new directions on the time integration of differential equations that include the development of Runge-Kutta methods that avoid order reduction and multistep methods for differential algebraic equations. The key research contributions will be (A) developing novel construction approaches and algebraic theory for explicit, and implicit-explicit Runge-Kutta methods satisfying weak stage order; (B) proofs of time-stepping barrier theorems for stiff problems; and (C) development thrusts for pathways of the developed methods into community software. The concepts will be employed to provide new methodologies in three particular fluid flow applications: (i) the dispersive shallow water equations; (ii) the incompressible Navier-Stokes equations; and (iii) advection-dominated problems, including hyperbolic conservation laws.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.