Note on two formulations of Crank-Nicolson method for Navier-Stokes equations

Alexander E. Labovsky; Kyle J. Schwiebert

doi:10.1007/s11075-024-01983-w

Note on two formulations of Crank-Nicolson method for Navier-Stokes equations

Alexander E. Labovsky, Kyle J. Schwiebert

Multiscale Methods

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Abstract

We consider two formulations of the Crank-Nicolson (CN) method for the Navier-Stokes equations (NSE). The “natural” way of implementing CN for NSE is formally second order accurate in time for both velocity and pressure, whereas another formulation approximates pressure with only first order accuracy in time. Both versions of the method are applied to the benchmark problem of computing drag and lift in the flow around a cylinder. We show that the presumably more accurate version of the CN can create a solution with nonphysical oscillations and give incorrect predictions for the maximal drag coefficient, whereas the other formulation of the method predicts the drag and lift coefficients more accurately and does not introduce nonphysical oscillations. We locate the source of the issue and suggest several remedies.

Original language	English
Journal	Numerical Algorithms
DOIs	https://doi.org/10.1007/s11075-024-01983-w
State	Accepted/In press - 2024

Funding

Notice: This manuscript has been authored in part by UT-Battelle, LLC under contract No. DE-AC05-00OR22725 with the U.S. Department of energy (DOE). The U.S. government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for U.S. government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan ( https://energy.gov/downloads/doe-public-access-plan ).

Keywords

Crank-Nicolson
Drag-lift coefficients
Navier-Stokes
Oscillations

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title = "Note on two formulations of Crank-Nicolson method for Navier-Stokes equations",

abstract = "We consider two formulations of the Crank-Nicolson (CN) method for the Navier-Stokes equations (NSE). The “natural” way of implementing CN for NSE is formally second order accurate in time for both velocity and pressure, whereas another formulation approximates pressure with only first order accuracy in time. Both versions of the method are applied to the benchmark problem of computing drag and lift in the flow around a cylinder. We show that the presumably more accurate version of the CN can create a solution with nonphysical oscillations and give incorrect predictions for the maximal drag coefficient, whereas the other formulation of the method predicts the drag and lift coefficients more accurately and does not introduce nonphysical oscillations. We locate the source of the issue and suggest several remedies.",

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AU - Labovsky, Alexander E.

AU - Schwiebert, Kyle J.

N1 - Publisher Copyright: © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

PY - 2024

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N2 - We consider two formulations of the Crank-Nicolson (CN) method for the Navier-Stokes equations (NSE). The “natural” way of implementing CN for NSE is formally second order accurate in time for both velocity and pressure, whereas another formulation approximates pressure with only first order accuracy in time. Both versions of the method are applied to the benchmark problem of computing drag and lift in the flow around a cylinder. We show that the presumably more accurate version of the CN can create a solution with nonphysical oscillations and give incorrect predictions for the maximal drag coefficient, whereas the other formulation of the method predicts the drag and lift coefficients more accurately and does not introduce nonphysical oscillations. We locate the source of the issue and suggest several remedies.

AB - We consider two formulations of the Crank-Nicolson (CN) method for the Navier-Stokes equations (NSE). The “natural” way of implementing CN for NSE is formally second order accurate in time for both velocity and pressure, whereas another formulation approximates pressure with only first order accuracy in time. Both versions of the method are applied to the benchmark problem of computing drag and lift in the flow around a cylinder. We show that the presumably more accurate version of the CN can create a solution with nonphysical oscillations and give incorrect predictions for the maximal drag coefficient, whereas the other formulation of the method predicts the drag and lift coefficients more accurately and does not introduce nonphysical oscillations. We locate the source of the issue and suggest several remedies.

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KW - Drag-lift coefficients

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KW - Oscillations

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SN - 1017-1398

JO - Numerical Algorithms

JF - Numerical Algorithms

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Note on two formulations of Crank-Nicolson method for Navier-Stokes equations

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