An Integral Equation Approach to the Incompressible Navier--Stokes Equations in Two Dimensions

We present a collection of methods for solving the incompressible Navier--Stokes equations in the plane that are based on a pure stream function formulation. The advantages of this approach are twofold: first, the velocity is automatically divergence free, and second, complicated (nonlocal) boundary conditions for the vorticity are avoided. The disadvantage is that the solution of a nonlinear fourth-order partial differential equation is required. By recasting this partial differential equation as an integral equation, we avoid the ill-conditioning which hampers finite difference and finite element methods in this environment. By using fast algorithms for the evaluation of volume integrals, we are able to solve the equations using O(M) or O(M log M) operations, where M is the number of points in the discretization of the domain.