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的时间演化方程。本项目旨在发展新的数学理论和方法，在保留现有方法的一般结构的同时，提高数值方法解决此类问题的准确性和效率。该研究包括适用于求解波动方程的方法，例如在声学或电磁学中出现的方程，以及具有约束的方程，例如在流体流动模拟中出现的方程。一个关键的应用将是增强浅水流体运动的模拟，它可以描述风暴潮、海啸和极端天气（如飓风）产生的波浪等现象。该项目将支持两名研究生，他们将由两所机构的教师共同指导，还包括本科生研究部分。本项目旨在建立微分方程时间积分的新方向，包括发展避免降阶的龙格-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)为满足弱阶段阶的显式和隐式-显式龙格-库塔方法开发新的构造方法和代数理论；(B)刚性问题时步势垒定理的证明；(C)将开发的方法引入社区软件的开发推力。这些概念将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；(3)以平流为主的问题，包括双曲守恒定律。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。

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.