Essentially compact schemes for unsteady viscous incompressible flows

E. Weinan, Jian Guo Liu

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A new fourth-order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced. The scheme is based on the vorticity-stream function formulation. It is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions. It is also very efficient, at every time step or Runge-Kutta stage, only two Poisson-like equations have to be solved. The Poisson-like equations are amenable to standard fast Poisson solvers usually designed for second order schemes. Detailed comparison with the second-order scheme shows the clear superiority of this new fourth-order scheme in resolving both the boundary layers and the gross features of the flow. This efficient fourth-order scheme also made it possible to compute the driven cavity flow at Reynolds number 106 on a 10242 grid at a reasonable cost. Fourth-order convergence is proved under mild regularity requirements. This is the first such result to our knowledge.

Original languageEnglish (US)
Pages (from-to)122-138
Number of pages17
JournalJournal of Computational Physics
Issue number1
StatePublished - Jun 1996

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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