### Abstract

This paper investigates different high order finite difference schemes and their accuracy for Burgers equation and Navier-Stokes equations. On a coarse grid, theoretical and numerical analysis indicate that a higher order difference scheme does not necessarily obtain more accurate solutions than a lower order scheme in the regions of high gradient variation (2nd order derivative). On a coarse grid, the numerical experiments indicate that the 3rd order biased upwind differencing for the inviscid fluxes with 2nd order central differencing for viscous term outperforms all other schemes with the accuracy order as high as 7th order. For the linear Burgers equation, both the analytical and numerical results indicate that a higher order scheme will be more accurate than a lower order scheme only when ReΔx ≤ 1. For the nonlinear Burgers equation, the numerical experiments also show the same conclusion. A numerical solution of laminar wall boundary layer from Navier-Stokes equations shows the same observation that a lower order scheme is more accurate on a coarse grid. The conception that a high order scheme can reduce mesh size unconditionally may be incorrect. Since the discrepancy occurs mostly in the region with high gradient variation, the refined mesh should not only be near the wall surface where the gradient is high, but also near the edge of the boundary layer where the gradient variation is high. The mesh spacing criterion for Navier-Stokes equation that a higher order scheme will be more accurate than a lower order scheme is not obtained yet. Using the criterion of the linear Burgers equation for Navier-Stokes equations may be excessive and preventively expensive. To resolve a laminar wall boundary layer accurately, the recommended grid Reynolds number is to be below 25 across the boundary layer. However, this is the criterion from the numerical experiments for all the finite differencing schemes, not for the higher order schemes to have better accuracy.

Original language | English |
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Title of host publication | 47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition |

State | Published - Dec 1 2009 |

Event | 47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition - Orlando, FL, United States Duration: Jan 5 2009 → Jan 8 2009 |

### Other

Other | 47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition |
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Country | United States |

City | Orlando, FL |

Period | 1/5/09 → 1/8/09 |

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### ASJC Scopus subject areas

- Space and Planetary Science
- Aerospace Engineering

### Cite this

*47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition*[2009-1137]

**High order finite differencing schemes and their accuracy for CFD.** / Shen, Yiqing; Zha, GeCheng.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition.*, 2009-1137, 47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, FL, United States, 1/5/09.

}

TY - GEN

T1 - High order finite differencing schemes and their accuracy for CFD

AU - Shen, Yiqing

AU - Zha, GeCheng

PY - 2009/12/1

Y1 - 2009/12/1

N2 - This paper investigates different high order finite difference schemes and their accuracy for Burgers equation and Navier-Stokes equations. On a coarse grid, theoretical and numerical analysis indicate that a higher order difference scheme does not necessarily obtain more accurate solutions than a lower order scheme in the regions of high gradient variation (2nd order derivative). On a coarse grid, the numerical experiments indicate that the 3rd order biased upwind differencing for the inviscid fluxes with 2nd order central differencing for viscous term outperforms all other schemes with the accuracy order as high as 7th order. For the linear Burgers equation, both the analytical and numerical results indicate that a higher order scheme will be more accurate than a lower order scheme only when ReΔx ≤ 1. For the nonlinear Burgers equation, the numerical experiments also show the same conclusion. A numerical solution of laminar wall boundary layer from Navier-Stokes equations shows the same observation that a lower order scheme is more accurate on a coarse grid. The conception that a high order scheme can reduce mesh size unconditionally may be incorrect. Since the discrepancy occurs mostly in the region with high gradient variation, the refined mesh should not only be near the wall surface where the gradient is high, but also near the edge of the boundary layer where the gradient variation is high. The mesh spacing criterion for Navier-Stokes equation that a higher order scheme will be more accurate than a lower order scheme is not obtained yet. Using the criterion of the linear Burgers equation for Navier-Stokes equations may be excessive and preventively expensive. To resolve a laminar wall boundary layer accurately, the recommended grid Reynolds number is to be below 25 across the boundary layer. However, this is the criterion from the numerical experiments for all the finite differencing schemes, not for the higher order schemes to have better accuracy.

AB - This paper investigates different high order finite difference schemes and their accuracy for Burgers equation and Navier-Stokes equations. On a coarse grid, theoretical and numerical analysis indicate that a higher order difference scheme does not necessarily obtain more accurate solutions than a lower order scheme in the regions of high gradient variation (2nd order derivative). On a coarse grid, the numerical experiments indicate that the 3rd order biased upwind differencing for the inviscid fluxes with 2nd order central differencing for viscous term outperforms all other schemes with the accuracy order as high as 7th order. For the linear Burgers equation, both the analytical and numerical results indicate that a higher order scheme will be more accurate than a lower order scheme only when ReΔx ≤ 1. For the nonlinear Burgers equation, the numerical experiments also show the same conclusion. A numerical solution of laminar wall boundary layer from Navier-Stokes equations shows the same observation that a lower order scheme is more accurate on a coarse grid. The conception that a high order scheme can reduce mesh size unconditionally may be incorrect. Since the discrepancy occurs mostly in the region with high gradient variation, the refined mesh should not only be near the wall surface where the gradient is high, but also near the edge of the boundary layer where the gradient variation is high. The mesh spacing criterion for Navier-Stokes equation that a higher order scheme will be more accurate than a lower order scheme is not obtained yet. Using the criterion of the linear Burgers equation for Navier-Stokes equations may be excessive and preventively expensive. To resolve a laminar wall boundary layer accurately, the recommended grid Reynolds number is to be below 25 across the boundary layer. However, this is the criterion from the numerical experiments for all the finite differencing schemes, not for the higher order schemes to have better accuracy.

UR - http://www.scopus.com/inward/record.url?scp=78549284957&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78549284957&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:78549284957

SN - 9781563479694

BT - 47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition

ER -