A new mixed potential representation for the equations of unsteady, incompressible flow

TL;DR

This paper introduces a novel integral representation for unsteady incompressible flow equations, combining heat and harmonic potentials to develop high-order accurate boundary integral equation solvers.

Contribution

It proposes a new mixed potential integral representation for unsteady flow equations, enabling well-conditioned coupled integral equations for velocity boundary conditions.

Findings

Well-conditioned integral equations for velocity components

High-order accuracy achieved with predictor-corrector and spectral deferred correction

Boundary densities as fundamental unknowns in the representation

Abstract

We present a new integral representation for the unsteady, incompressible Stokes or Navier-Stokes equations, based on a linear combination of heat and harmonic potentials. For velocity boundary conditions, this leads to a coupled system of integral equations: one for the normal component of velocity and one for the tangential components. Each individual equation is well-condtioned, and we show that using them in predictor-corrector fashion, combined with spectral deferred correction, leads to high-order accuracy solvers. The fundamental unknowns in the mixed potential representation are densities supported on the boundary of the domain. We refer to one as the vortex source, the other as the pressure source and the coupled system as the combined source integral equation.