Collaborative Research: Variational Structure Preserving Methods for Incompressible Flows: Discretization, Analysis, and Parallel Solvers

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The goal of this project is to develop improved models and numerical methods that advance the state of the art for incompressible fluid flow simulation. The key ideas are the better enforcement of geometric and physical laws in the computer algorithms, building a solid mathematical framework for the models and methods developed, and the devising of algorithms that will allow for efficient implementation on supercomputers. Although the simulation of fluid flow is a critical subtask in a wide spectrum of engineering applications, current tools and techniques are often unreliable and it is not uncommon for state of the art methods to take weeks or months (or possibly never finish with an accurate solution), even with thousands of processors, to perform simulations of flows around a car, through a nuclear reactor, or around part of an airplane. The project aims to develop mathematical prediction models and numerical methods that will provide more accurate solutions in a more efficient manner than state-of-the-art methods.The PIs will construct efficient methods for incompressible flow simulation by constructing models and methods that better adhere to geometric structure and physical conservation laws than modern methods. The key components are to i) construct novel methods that are efficient and can correctly account for vorticity dynamics, and energy, helicity, and mass conservation -- this will require development of efficient boundary conditions and significant analysis to build a solid mathematical framework; ii) develop more efficient algebraic solvers for these methods that can be used on thousands of processors; iii) large scale testing on benchmark problems as well as on application problems with collaborators. Broader impacts include i) developing efficient methods for simulating high speed incompressible flows, which will improve the design process for a wide spectrum of applications in environmental engineering, in cardiovascular simulations, and in atmosphere and ocean sciences; ii) training graduate and undergraduate students through research involvement; and iii) developing large scale, parallel codes to be made publicly available as part of the deal.II library.