## 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:

- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing
- Tulane Univ., New Orleans, LA (United States). Dept. of Mathematics
- 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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