A new mixed potential representation for unsteady, incompressible flow

Leslie Greengard, Shidong Jiang

Research output: Contribution to journalReview articlepeer-review


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-conditioned, 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 to the coupled system as the combined source integral equation.

Original languageEnglish (US)
Pages (from-to)733-755
Number of pages23
JournalSIAM Review
Issue number4
StatePublished - 2019


  • Boundary integral equations
  • Harmonic potentials
  • Heat potentials
  • Mixed potential formulation
  • Navier-Stokes equations
  • Predictor-corrector method
  • Spectral deferred correction method
  • Unsteady Stokes ow

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Mathematics
  • Applied Mathematics


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