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

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.