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TECHNICAL PAPERS

Limitations of Richardson Extrapolation and Some Possible Remedies

[+] Author and Article Information
Ismail Celik1

Mechanical and Aerospace Engineering Department, West Virginia University, Morgantown, WV 26506-6106

Jun Li, Gusheng Hu, Christian Shaffer

Mechanical and Aerospace Engineering Department, West Virginia University, Morgantown, WV 26506-6106

1

Please send all communications to the first author at Ismail.Celik@mail.wvu.edu

J. Fluids Eng 127(4), 795-805 (Apr 28, 2005) (11 pages) doi:10.1115/1.1949646 History: Received August 27, 2004; Revised April 27, 2005; Accepted April 28, 2005

The origin of oscillatory convergence in finite difference methods is investigated. Fairly simple implicit schemes are used to solve the steady one-dimensional convection-diffusion equation with variable coefficients, and possible scenarios are shown that exhibit the oscillatory convergence. Also, a manufactured solution to difference equations is formulated that exhibits desired oscillatory behavior in grid convergence, with a varying formal order of accuracy. This model-error equation is used to statistically assess the performance of several methods of extrapolation. Alternative extrapolation schemes, such as the deferred extrapolation to limit technique, to calculate the coefficients in the Taylor series expansion of the error function are also considered. A new method is proposed that is based on the extrapolation of approximate error, and is shown to be a viable alternative to other methods. This paper elucidates the problem of oscillatory convergence, and brings a new perspective to the problem of estimating discretization error by optimizing the information from a minimum number of calculations.

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Copyright © 2005 by American Society of Mechanical Engineers
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References

Figures

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Figure 2

Normalized streamwise velocity difference based on the medium grid at x∕H=1 in a 2D turbulent flow over a backward-facing step (using FLUENT and Spalart-Allmaras turbulence model (11))

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Figure 7

Error plots showing cases II(a) and II(b) in both expanded and zoomed form; the lower plots show convergence behavior for very small dx values

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Figure 9

Error plots showing cases IV(a) and IV(b) in both expanded and zoomed form; the lower plots show convergence behavior for very small dx values

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Figure 10

Solutions for cases I—IV using upwinding and central differencing schemes, as well as the semi-analytical solution

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Figure 11

Difference between analytical and numerical solutions at x=0.5 for equation 10

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Figure 1

An example of oscillatory convergence. First-order upwinding scheme applied to the convection term and second-order central differencing applied to the diffusion term with u=cos(4πx);Γ is the diffusion coefficient. (see Appendix)

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Figure 3

Designed solutions ϕe to Eq. 38 and numerical solutions ϕn to Eq. 35 with the manufactured scheme

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Figure 4

An example of the behavior of oscillatory model Eq. 41

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Figure 5

Probability of ϕ(0) at different intervals, predicted with different methods

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Figure 6

Error plots showing cases I(a) and I(b) in both expanded and zoomed form

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Figure 8

Error plots showing cases III(a) and III(b) in both expanded and zoomed form; the lower plots show convergence behavior for very small dx values

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