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

基于物理或工程的现实模型的仿真通常需要求解大型时间演化方程。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，从而将这些问题时间推进，同时保留已建立方法的一般结构。该研究包括适用于求解波动方程的方法，例如声学或电磁学中产生的波动方程，以及具有约束条件的方程，例如流体流动模拟中产生的方程。一个关键的应用将是浅水流体运动的增强模拟，它可以描述诸如风暴潮、海啸和通过极端天气（例如，飓风）。该项目将资助两名研究生，他们将由两个机构的教师共同指导，还包括本科生研究部分。该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Runge-Kutta方法和微分代数方程的多步法。主要的研究贡献将是（A）开发新的构造方法和代数理论的显式，和隐式显式龙格库塔方法满足弱阶段顺序;（B）证明时间步进障碍定理的刚性问题;和（C）开发推力的路径开发的方法进入社区软件。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;（iii）对流主导的问题，包括双曲守恒律。这个奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.