[hityangsir]

Hi All,

I am confused with a basic problem of CFD.

When we derive the discretized formula with upwind central differencing or other scheme, the numerical dissipation and dispersion terms are used to evaluate the accuracy and stability of the specific scheme. I can understand how these terms are derived. However, the physical and numerical meaning of these terms are confusing? Why are they called numerical diffusion and dispersion? And what is their effect on accuracy and stability?

Thank you

Richard

[FMDenaro]

Originally Posted by hityangsir

Hi All,

I am confused with a basic problem of CFD.

When we derive the discretized formula with upwind central differencing or other scheme, the numerical dissipation and dispersion terms are used to evaluate the accuracy and stability of the specific scheme. I can understand how these terms are derived. However, the physical and numerical meaning of these terms are confusing? Why are they called numerical diffusion and dispersion? And what is their effect on accuracy and stability?

Thank you

Richard

consider the equation

df/dt + u df/dx =0

u positive constant

assume f(x,0)=f0(x) a periodic wave function.
The exact solution is a travelling wave at constant speed u. Numerical diffusion is responsible of the fact that the peaks (min and max) of the wave are not maintaned but diminuish in time. Numerical dispersion is responsible of the fact that packet of waves travel ad speed different from u.

You cna use Fourier analysis to analyse the effects

[Bazinga]

Originally Posted by hityangsir

Hi All,

I am confused with a basic problem of CFD.

When we derive the discretized formula with upwind central differencing or other scheme, the numerical dissipation and dispersion terms are used to evaluate the accuracy and stability of the specific scheme. I can understand how these terms are derived. However, the physical and numerical meaning of these terms are confusing? Why are they called numerical diffusion and dispersion? And what is their effect on accuracy and stability?

Thank you

Richard

I think the truncated part of the Taylor expansion is used to evaluate the order of the scheme (accuracy). The truncation error consists of numerical diffusion terms (even orders) and numerical dispersion terms (odd orders). Both have no physical meaning, but are numerical phenomena that act like physical phenomena. Numerical diffusion blurrs the values while numerical dispersion adds wiggles into your solution. Note that the second order term of the Taylor expansion looks like a diffusive term of the transport equations.

It seems like lower order numerical schemes are more stable due to the numerical phenomena described above. Numerical schemes of high accuracy sometimes introduce an additional artificial diffusion term to help stablilize those schemes.

Diffusion and dissipation are sometimes used interchangeable and sometimes the even order terms are called dissipation and then the combination of dissipation and dispersion is named diffusion.

I am happy if somebody can provide reference to a text book or expand on my explanations because I am also not sure if my explanation is correct.

[FMDenaro]

this is a classical topic of CFD, there are many well known textbooks (older or not) that explain the matter.

[robo]

Originally Posted by Bazinga

I think the truncated part of the Taylor expansion is used to evaluate the order of the scheme (accuracy).

The order of the scheme does NOT tell you the accuracy; it tells you the rate at which the accuracy improves with successive grid refinement. The accuracy can only be measured by validation studies; nothing about the numerics of an isolated solution (ie only a single solution, no experimental data) can tell you about the accuracy.

My preferred references for numerical diffusion/dispersion are the textbooks computational fluid mechanics & heat transfer by Pletcher, Tannehill, and Anderson, and computational aeroacoustics by Tam. The acoustic one will emphasize spectral characteristics a lot more, which may not be useful to everyone.

[Radwanma]

Hi all;

How can I change the magnitude of the Y+ (wall distance ) in the ICEM ???

Regards,

[Bazinga]

Originally Posted by robo

The order of the scheme does NOT tell you the accuracy; it tells you the rate at which the accuracy improves with successive grid refinement. The accuracy can only be measured by validation studies; nothing about the numerics of an isolated solution (ie only a single solution, no experimental data) can tell you about the accuracy.

My preferred references for numerical diffusion/dispersion are the textbooks computational fluid mechanics & heat transfer by Pletcher, Tannehill, and Anderson, and computational aeroacoustics by Tam. The acoustic one will emphasize spectral characteristics a lot more, which may not be useful to everyone.

Dear robo,

thank you very much for correcting my mistake and additionally providing the literature. You are correct! :-)

I will study your suggested textbooks.

Best regards

[hityangsir]

Dear Filippo,
Thank you for your kind reply. This is very clear understanding

[hityangsir]

Originally Posted by robo

The order of the scheme does NOT tell you the accuracy; it tells you the rate at which the accuracy improves with successive grid refinement. The accuracy can only be measured by validation studies; nothing about the numerics of an isolated solution (ie only a single solution, no experimental data) can tell you about the accuracy.

My preferred references for numerical diffusion/dispersion are the textbooks computational fluid mechanics & heat transfer by Pletcher, Tannehill, and Anderson, and computational aeroacoustics by Tam. The acoustic one will emphasize spectral characteristics a lot more, which may not be useful to everyone.

You are right. Accuracy of CFD is such a broad and open question, which is essential but difficult to achieve

[hityangsir]

Originally Posted by Bazinga

Dear robo,

thank you very much for correcting my mistake and additionally providing the literature. You are correct! :-)

I will study your suggested textbooks.

Best regards

Dear Bazinga
Thank you for your reply.
I have collect some material that may help:
http://elearning.cerfacs.fr/pdfs/numerical/FDscheme.pdf
http://www.mathematik.uni-dortmund.d.../lecture10.pdf
https://people.maths.ox.ac.uk/trefethen/5all.pdf

Regards