A fast lattice Green's function method for solving viscous incompressible flows on unbounded domains

TL;DR

This paper introduces a computationally efficient lattice Green's function method for simulating three-dimensional viscous incompressible flows on unbounded domains, combining advanced numerical techniques for accuracy and scalability.

Contribution

It presents a novel lattice Green's function approach that limits operations to a finite domain, enabling fast, scalable, and accurate simulations of unbounded viscous flows.

Findings

Achieves linear computational complexity.

Demonstrates high accuracy with vortex ring simulations.

Enables parallel scalable computations.

Abstract

A computationally efficient method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. The method formally discretizes the incompressible Navier-Stokes equations on an unbounded staggered Cartesian grid. Operations are limited to a finite computational domain through a lattice Green's function technique. This technique obtains solutions to inhomogeneous difference equations through the discrete convolution of source terms with the fundamental solutions of the discrete operators. The differential algebraic equations describing the temporal evolution of the discrete momentum equation and incompressibility constraint are numerically solved by combining an integrating factor technique for the viscous term and a half-explicit Runge-Kutta scheme for the convective term. A projection method that exploits the mimetic and commutativity properties of the discrete operators is used to efficiently solve the system of equations that arises in each stage of the time integration scheme. Linear complexity, fast computation rates, and parallel scalability are achieved using recently developed fast multipole methods for difference equations. The accuracy and physical fidelity of solutions is verified through numerical simulations of vortex rings.