Accurate Projection Methods for the Incompressible Navier–Stokes Equations

Brown, David L.; Cortez, Ricardo; Minion, Michael L.

doi:10.1006/jcph.2001.6715

Accurate Projection Methods for the Incompressible Navier–Stokes Equations

Journal Article · Tue Apr 10 00:00:00 EDT 2001 · Journal of Computational Physics

DOI:https://doi.org/10.1006/jcph.2001.6715· OSTI ID:1429358

Brown, David L. ^[1]; Cortez, Ricardo ^[2]; Minion, Michael L. ^[3]

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

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.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States); Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); National Science Foundation (NSF); Alfred P. Sloan Foundation

Grant/Contract Number:: AC02-05CH11231; FG02-92ER25139; W-7405-ENG-48; W-7405-ENG-36

OSTI ID:: 1429358

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: 2 Vol. 168; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (34)

An improved projection scheme applied to pseudospectral methods for the incompressible Navier-Stokes equations Hugues, Sandrine; Randriamampianina, Anthony International Journal for Numerical Methods in Fluids, Vol. 28, Issue 3 https://doi.org/10.1002/(SICI)1097-0363(19980915)28:3<501::AID-FLD730>3.0.CO;2-S	journal	September 1998
An Analysis of the Fractional Step Method Perot, J. Blair Journal of Computational Physics, Vol. 108, Issue 1 https://doi.org/10.1006/jcph.1993.1162	journal	September 1993
A Fourth-Order Accurate Method for the Incompressible Navier-Stokes Equations on Overlapping Grids Henshaw, William D. Journal of Computational Physics, Vol. 113, Issue 1 https://doi.org/10.1006/jcph.1994.1114	journal	July 1994
An Impulse-Based Approximation of Fluid Motion due to Boundary Forces Cortez, Ricardo Journal of Computational Physics, Vol. 123, Issue 2 https://doi.org/10.1006/jcph.1996.0028	journal	February 1996
A Finite-Difference Scheme for Three-Dimensional Incompressible Flows in Cylindrical Coordinates Verzicco, R.; Orlandi, P. Journal of Computational Physics, Vol. 123, Issue 2 https://doi.org/10.1006/jcph.1996.0033	journal	February 1996
On the Stability of Godunov-Projection Methods for Incompressible Flow Minion, Michael L. Journal of Computational Physics, Vol. 123, Issue 2 https://doi.org/10.1006/jcph.1996.0035	journal	February 1996
A Projection Method for Locally Refined Grids Minion, Michael L. Journal of Computational Physics, Vol. 127, Issue 1 https://doi.org/10.1006/jcph.1996.0166	journal	August 1996
Finite Difference Schemes for Incompressible Flows in the Velocity–Impulse Density Formulation E., Weinan; Liu, Jian-Guo Journal of Computational Physics, Vol. 130, Issue 1 https://doi.org/10.1006/jcph.1996.5537	journal	January 1997
Velicity Methods: Lagrangian Numerical Methods which Preserve the Hamiltonian Structure of Incompressible Fluid Flow Buttke, Tomas F. Vortex Flows and Related Numerical Methods https://doi.org/10.1007/978-94-015-8137-0_3	book	January 1993
Remark on the pressure boundary condition for the projection method Temam, R. Theoretical and Computational Fluid Dynamics, Vol. 3, Issue 3 https://doi.org/10.1007/BF00271801	journal	December 1991
Boundary conditions for incompressible flows Orszag, Steven A.; Israeli, Moshe; Deville, Michel O. Journal of Scientific Computing, Vol. 1, Issue 1 https://doi.org/10.1007/BF01061454	journal	January 1986
A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows Goda, Katuhiko Journal of Computational Physics, Vol. 30, Issue 1 https://doi.org/10.1016/0021-9991(79)90088-3	journal	January 1979
Application of a fractional-step method to incompressible Navier-Stokes equations Kim, J.; Moin, P. Journal of Computational Physics, Vol. 59, Issue 2 https://doi.org/10.1016/0021-9991(85)90148-2	journal	June 1985
A second-order projection method for the incompressible navier-stokes equations Bell, John B.; Colella, Phillip; Glaz, Harland M. Journal of Computational Physics, Vol. 85, Issue 2 https://doi.org/10.1016/0021-9991(89)90151-4	journal	December 1989
High-order splitting methods for the incompressible Navier-Stokes equations Karniadakis, George Em; Israeli, Moshe; Orszag, Steven A. Journal of Computational Physics, Vol. 97, Issue 2 https://doi.org/10.1016/0021-9991(91)90007-8	journal	December 1991
A fourth-order-accurate difference approximation for the incompressible Navier-Stokes equations Henshaw, William D.; Kreiss, Heinz-Otto; Reyna, Luis G. M. Computers & Fluids, Vol. 23, Issue 4 https://doi.org/10.1016/0045-7930(94)90053-1	journal	May 1994
On the solution of the Navier-Stokes equations using Chebyshev projection schemes with third-order accuracy in time Botella, Olivier Computers & Fluids, Vol. 26, Issue 2 https://doi.org/10.1016/S0045-7930(96)00032-1	journal	February 1997
Numerical vorticity creation based on impulse conservation. Summers, D. M.; Chorin, A. J. Proceedings of the National Academy of Sciences, Vol. 93, Issue 5 https://doi.org/10.1073/pnas.93.5.1881	journal	March 1996
Numerical simulation of incompressible viscous flow in deforming domains Colella, P.; Trebotich, D. P. Proceedings of the National Academy of Sciences, Vol. 96, Issue 10 https://doi.org/10.1073/pnas.96.10.5378	journal	May 1999
An Adaptive Projection Method for Unsteady, Low-Mach Number Combustion Pember, R. B.; Howell, L. H.; Bell, J. B. Combustion Science and Technology, Vol. 140, Issue 1-6 https://doi.org/10.1080/00102209808915770	journal	December 1998
Numerical solution of the Navier-Stokes equations Chorin, Alexandre Joel Mathematics of Computation, Vol. 22, Issue 104 https://doi.org/10.1090/S0025-5718-1968-0242392-2	journal	January 1968
On the convergence of discrete approximations to the Navier-Stokes equations Chorin, Alexandre Joel Mathematics of Computation, Vol. 23, Issue 106 https://doi.org/10.1090/S0025-5718-1969-0242393-5	journal	May 1969
On error estimates of the projection methods for the Navier-Stokes equations: Second-order schemes Shen, Jie Mathematics of Computation, Vol. 65, Issue 215 https://doi.org/10.1090/S0025-5718-96-00750-8	journal	July 1996
Projection Method I: Convergence and Numerical Boundary Layers E., Weinan; Liu, Jian-Guo SIAM Journal on Numerical Analysis, Vol. 32, Issue 4 https://doi.org/10.1137/0732047	journal	August 1995
A Second-Order Accurate Pressure-Correction Scheme for Viscous Incompressible Flow van Kan, J. SIAM Journal on Scientific and Statistical Computing, Vol. 7, Issue 3 https://doi.org/10.1137/0907059	journal	July 1986
Projection Method II: Godunov–Ryabenki Analysis E., Weinan; Liu, Jian-Guo SIAM Journal on Numerical Analysis, Vol. 33, Issue 4 https://doi.org/10.1137/S003614299426450X	journal	August 1996
The Accuracy of the Fractional Step Method Strikwerda, John C.; Lee, Young S. SIAM Journal on Numerical Analysis, Vol. 37, Issue 1 https://doi.org/10.1137/S0036142997326938	journal	January 1999
A Numerical Method for the Incompressible Navier-Stokes Equations Based on an Approximate Projection Almgren, Ann S.; Bell, John B.; Szymczak, William G. SIAM Journal on Scientific Computing, Vol. 17, Issue 2 https://doi.org/10.1137/S1064827593244213	journal	March 1996
Hamilton-based Numerical Methods for a Fluid-Membrane Interaction in Two and Three Dimensions Recchioni, Maria Cristina; Russo, Giovanni SIAM Journal on Scientific Computing, Vol. 19, Issue 3 https://doi.org/10.1137/S106482759528973X	journal	May 1998
Approximate Projection Methods: Part I. Inviscid Analysis Almgren, Ann S.; Bell, John B.; Crutchfield, William Y. SIAM Journal on Scientific Computing, Vol. 22, Issue 4 https://doi.org/10.1137/S1064827599357024	journal	January 2000
Impulse-based methods for fluid flow Cortez, Ricardo https://doi.org/10.2172/87798	dissertation	May 1995
An efficient second-order projection method for viscous incompressible flow Bell, John; Howell, Louis; Colella, Phillip 10th Computational Fluid Dynamics Conference https://doi.org/10.2514/6.1991-1560	conference	February 2013
An adaptive projection method for the incompressible Euler equations Almgren, Ann; Bell, John; Colella, Phillip 11th Computational Fluid Dynamics Conference https://doi.org/10.2514/6.1993-3345	conference	August 1993
A projection method for combustion in the zero Mach number limit Lai, Mindy; Bell, John; Colella, Philip 11th Computational Fluid Dynamics Conference https://doi.org/10.2514/6.1993-3369	conference	August 1993

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