Accurate Projection Methods for the Incompressible Navier–Stokes Equations

doi:10.1006/jcph.2001.6715

Journal of Computational Physics

Volume 168, Issue 2, 10 April 2001, Pages 464-499

https://doi.org/10.1006/jcph.2001.6715 Get rights and content

Abstract

This paper considers the accuracy of projection method approximations to the initial–boundary-value problem for the incompressible Navier–Stokes equations. The issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second-order methodology a decade and a half ago. It has been observed that while the velocity can be reliably computed to second-order accuracy in time and space, the pressure is typically only first-order accurate in the L_∞-norm. This paper identifies the source of this problem in the interplay of the global pressure-update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second-order accurate, as demonstrated by a normal mode analysis and numerical experiments. In addition, a numerical method based on a gauge variable formulation of the incompressible Navier–Stokes equations, which provides another option for obtaining fully second-order convergence in both velocity and pressure, is discussed. The connection between the boundary conditions for projection methods and the gauge method is explained in detail.

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^f1: [email protected]

^f2: [email protected]

^f3: [email protected]

¹: The work of this author was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory and Los Alamos National Laboratory under Contracts W-7405-ENG-48 and W-7405-ENG-36.

²: Supported in part by NSF Grant DMS-9816951.

³: The work of this author was performed in part under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory and Los Alamos National Laboratory under Contracts W-7405-ENG-48 and W-7405-ENG-36. Support also provided by the U.S. Department of Energy under Contract DE-FG02-92ER25139, NSF Grant DMS-9973290, and the Alfred P. Sloan Foundation.

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Regular ArticleAccurate Projection Methods for the Incompressible Navier–Stokes Equations

Abstract

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A numerical method for the incompressible Navier–Stokes equations based on an approximate projection

SIAM J. Sci. Comput.

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Regular Article
Accurate Projection Methods for the Incompressible Navier–Stokes Equations