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Title: Accurate Projection Methods for the Incompressible Navier–Stokes Equations

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. Here, we identify 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.

Authors:
 [1];  [2];  [3]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing
  2. Tulane Univ., New Orleans, LA (United States). Dept. of Mathematics
  3. Univ. of North Carolina, Chapel Hill, NC (United States). Dept. of Mathematics
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); National Science Foundation (NSF); Alfred P. Sloan Foundation
OSTI Identifier:
1429358
Grant/Contract Number:  
AC02-05CH11231; FG02-92ER25139; W-7405-ENG-48; W-7405-ENG-36; DMS-9816951; DMS-9973290
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 168; Journal Issue: 2; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; incompressible flow; projection method; boundary conditions

Citation Formats

Brown, David L., Cortez, Ricardo, and Minion, Michael L. Accurate Projection Methods for the Incompressible Navier–Stokes Equations. United States: N. p., 2001. Web. doi:10.1006/jcph.2001.6715.
Brown, David L., Cortez, Ricardo, & Minion, Michael L. Accurate Projection Methods for the Incompressible Navier–Stokes Equations. United States. doi:10.1006/jcph.2001.6715.
Brown, David L., Cortez, Ricardo, and Minion, Michael L. Tue . "Accurate Projection Methods for the Incompressible Navier–Stokes Equations". United States. doi:10.1006/jcph.2001.6715. https://www.osti.gov/servlets/purl/1429358.
@article{osti_1429358,
title = {Accurate Projection Methods for the Incompressible Navier–Stokes Equations},
author = {Brown, David L. and Cortez, Ricardo and Minion, Michael L.},
abstractNote = {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. Here, we identify 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.},
doi = {10.1006/jcph.2001.6715},
journal = {Journal of Computational Physics},
number = 2,
volume = 168,
place = {United States},
year = {2001},
month = {4}
}

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