A Fast Integral Equation Method for the Two-Dimensional Navier-Stokes Equations

TL;DR

This paper presents a comprehensive and efficient integral equation-based solver for the two-dimensional Navier-Stokes equations, demonstrating its robustness and applicability to complex geometries with improved numerical properties.

Contribution

It introduces a complete integral equation method for Navier-Stokes, combining recent singular quadrature techniques with established numerical methods for complex domain flows.

Findings

Demonstrates convergence on various geometries

Shows improved computational performance

Validates robustness of the integral equation approach

Abstract

The integral equation approach to partial differential equations (PDEs) provides significant advantages in the numerical solution of the incompressible Navier-Stokes equations. In particular, the divergence-free condition and boundary conditions are handled naturally, and the ill-conditioning caused by high order terms in the PDE is preconditioned analytically. Despite these advantages, the adoption of integral equation methods has been slow due to a number of difficulties in their implementation. This work describes a complete integral equation-based flow solver that builds on recently developed methods for singular quadrature and the solution of PDEs on complex domains, in combination with several more well-established numerical methods. We apply this solver to flow problems on a number of geometries, both simple and challenging, studying its convergence properties and computational performance. This serves as a demonstration that it is now relatively straightforward to develop a robust, efficient, and flexible Navier-Stokes solver, using integral equation methods.