Central Differencing Scheme and False Diffusion

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 August 5, 2019, 08:02 Central Differencing Scheme and False Diffusion #1 Member   Raphael Join Date: Nov 2012 Posts: 68 Rep Power: 13 I have a two part question: (1) Does central difference scheme eliminate false diffusion (by false diffusion, i mean "diffusion" that is caused by misalignment with the grid, not mere discretization error)? (2) Central difference scheme produces oscillations and and is unstable with gauss-seidal/jacobi solvers. However, using either direct matrix inversion or differed correction approach, the equations can still be solved. Oscillations can be removed with flux limiter (just like upwind higher order schemes). Thus, why dont more people/software use central difference? In simple test cases of mine, it seems to completely eliminate false diffusion. It also appears completely monotonic for Pe = infinity.

August 5, 2019, 08:32
#2
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Filippo Maria Denaro
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Quote:
 Originally Posted by arkie87 I have a two part question: (1) Does central difference scheme eliminate false diffusion (by false diffusion, i mean "diffusion" that is caused by misalignment with the grid, not mere discretization error)? (2) Central difference scheme produces oscillations and and is unstable with gauss-seidal/jacobi solvers. However, using either direct matrix inversion or differed correction approach, the equations can still be solved. Oscillations can be removed with flux limiter (just like upwind higher order schemes). Thus, why dont more people/software use central difference? In simple test cases of mine, it seems to completely eliminate false diffusion. It also appears completely monotonic for Pe = infinity.

1) Central difference that has no artificial diffusion terms means you have an aligned and spatially balanced stencil (uniform grid). But just if you write a scheme on a non uniform grid with a "centred" number of values but on non balanced grid sizes, you see further effects.

2) who told you that central schemes are not used? If a central scheme is used but with a non linear scheme, you can have monotonic solutions, this is what the Godunov theorem says.

August 5, 2019, 08:46
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Raphael
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Quote:
 Originally Posted by FMDenaro 1) Central difference that has no artificial diffusion terms means you have an aligned and spatially balanced stencil (uniform grid). But just if you write a scheme on a non uniform grid with a "centred" number of values but on non balanced grid sizes, you see further effects. 2) who told you that central schemes are not used? If a central scheme is used but with a non linear scheme, you can have monotonic solutions, this is what the Godunov theorem says.
(1) my grid in my test problem is 45 degrees misaligned with the flow. So you are saying that if i use a uniform grid (in x and y), then central difference produces no artificial diffusion, and is, therefore, ideal?

(2) it is not available in Fluent for laminar problems?

August 5, 2019, 08:53
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Raphael
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Quote:
 Originally Posted by FMDenaro 1) If a central scheme is used but with a non linear scheme, you can have monotonic solutions, this is what the Godunov theorem says.
What do you mean by "non-linear" scheme? you mean flux limiter?

August 5, 2019, 09:07
#5
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Filippo Maria Denaro
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Quote:
 Originally Posted by arkie87 (1) my grid in my test problem is 45 degrees misaligned with the flow. So you are saying that if i use a uniform grid (in x and y), then central difference produces no artificial diffusion, and is, therefore, ideal? (2) it is not available in Fluent for laminar problems?

- yes, in the ideal case central discretization has no appearence of artificial diffusion/dissipation terms in the local truncation error.

- You can set central unbounded scheme in Fluent

August 5, 2019, 09:09
#6
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Filippo Maria Denaro
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Quote:
 Originally Posted by arkie87 What do you mean by "non-linear" scheme? you mean flux limiter?

the flux limiting procedure is one of the way to make non-linear a numerical scheme also if you solve a linear advection equation. This way, the limit of the Godunov theorem is elimninated

August 5, 2019, 09:17
#7
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Raphael
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Quote:
 Originally Posted by FMDenaro - yes, in the ideal case central discretization has no appearence of artificial diffusion/dissipation terms in the local truncation error. - You can set central unbounded scheme in Fluent
(1) ok. interesting. do you have any reference on this limiting case for further reading?

(2) does that mean fluent doesnt use flux limiter to keep central difference scheme bounded/monotonic?

August 5, 2019, 09:20
#8
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Raphael
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Quote:
 Originally Posted by FMDenaro the flux limiting procedure is one of the way to make non-linear a numerical scheme also if you solve a linear advection equation. This way, the limit of the Godunov theorem is elimninated
sorry, i am having a bit of trouble understanding you.

Are you saying that a linear advection equation needs to be made non-linear somehow (e.g. using a flux limiter) in order to allow for a non-oscillatory solution when solving using central difference (or any higher order scheme)?

Thanks!

August 5, 2019, 09:22
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Filippo Maria Denaro
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Quote:
 Originally Posted by arkie87 sorry, i am having a bit of trouble understanding you. Are you saying that a linear advection equation needs to be made non-linear somehow (e.g. using a flux limiter) in order to allow for a non-oscillatory solution when solving using central difference (or any higher order scheme)? Thanks!

Yes, the Godunov theorem states that only first order accurate linear scheme are monotonic. So, in order to get higher accuracy and mononotne solution, you have to make non-linear the numerical scheme even if the PDE is linear.

August 5, 2019, 09:27
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Filippo Maria Denaro
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Quote:
 Originally Posted by arkie87 (1) ok. interesting. do you have any reference on this limiting case for further reading? (2) does that mean fluent doesnt use flux limiter to keep central difference scheme bounded/monotonic?

1) this is a standard proof of any numerical method textbook, just have a look to the expression of the local truncation error

2) Fluent can work letting you setting one of both options, bounded or unbounded central scheme. This central discretization is available under the LES formulatiom

August 5, 2019, 09:35
#11
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Raphael
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Quote:
 Originally Posted by FMDenaro 1) this is a standard proof of any numerical method textbook, just have a look to the expression of the local truncation error 2) Fluent can work letting you setting one of both options, bounded or unbounded central scheme. This central discretization is available under the LES formulatiom

(1) I have ferziger's book, but i dont remember seeing something like this. And my understand is numerical diffusion and truncation error are very different. First order upwind has zero numerical diffusion if flow is aligned with grid, but non-zero truncation error.

(2) Is it available under standard laminar solver with a flux limiter? I know it isnt by default... why isnt it more popular to use, vs. 2nd order upwind (which is default scheme in fluent), which also needs a flux limiter etc... but has more false diffusion? I suppose if there is only zero false diffusion for limiting case of uniform grid central difference, then default should be more robust/widely applicable?

August 5, 2019, 09:43
#12
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Filippo Maria Denaro
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Quote:
 Originally Posted by arkie87 (1) I have ferziger's book, but i dont remember seeing something like this. And my understand is numerical diffusion and truncation error are very different. First order upwind has zero numerical diffusion if flow is aligned with grid, but non-zero truncation error. (2) Is it available under standard laminar solver with a flux limiter? I know it isnt by default... why isnt it more popular to use, vs. 2nd order upwind (which is default scheme in fluent), which also needs a flux limiter etc... but has more false diffusion? I suppose if there is only zero false diffusion for limiting case of uniform grid central difference, then default should be more robust/widely applicable?

1-no, you are wrong, the local truncation error is the key to analsye the terms that a certain discretization adds to the equation. When we talk about "artificial viscosity" we are considering the modified equation (original PDE + truncation error) where spurious terms appears in the form of additional diffusion. Upwind introduces artificial diffusion also on aligned grid. This is not the case of central scheme

2- because when using central discretization the code is less "robust" and is necessary that the user has a CFD background to have a full control of the solution.

August 5, 2019, 09:52
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Raphael
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Quote:
 Originally Posted by FMDenaro 1-no, you are wrong, the local truncation error is the key to analsye the terms that a certain discretization adds to the equation. When we talk about "artificial viscosity" we are considering the modified equation (original PDE + truncation error) where spurious terms appears in the form of additional diffusion. Upwind introduces artificial diffusion also on aligned grid. This is not the case of central scheme
i dont think i am wrong. i have read numerous papers on this subject, and have seen the taylor series expansions that show the "false diffusion" terms that are being added to the equation.

but as all those papers point out, False Diffusion" (as it is referred to in the literature) is a uniquely multidimensional phenomena. In 1-D advection equation, there is also a "false diffusion" in upwind scheme, which acts as a diffusive term, but this "false diffusion" is not usually what is meant by the term False Diffusion.

If you go by truncation error alone, then 2nd order upwind should also not have any false diffusion, since there is no term proportional to the 2nd derivative of PHI.

August 5, 2019, 09:55
#14
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Raphael
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Quote:
 Originally Posted by FMDenaro 1-no, you are wrong, the local truncation error is the key to analsye the terms that a certain discretization adds to the equation. When we talk about "artificial viscosity" we are considering the modified equation (original PDE + truncation error) where spurious terms appears in the form of additional diffusion. Upwind introduces artificial diffusion also on aligned grid. This is not the case of central scheme 2- because when using central discretization the code is less "robust" and is necessary that the user has a CFD background to have a full control of the solution.
i should also mention that in the literature, they distinguish between streamwise and cross-streamline diffusion. Streamwise false diffusion is what you are talking about (and is related to truncation error terms added from the scheme), whereas cross-streamline false diffusion is what is usually meant by "False Diffusion" as discussed in the literature.

Regardless of the semantics, i havent seen anything that indicates that there should be zero cross-streamline False Diffusion for central difference scheme on uniform grid.

August 5, 2019, 11:08
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Lucky
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Central schemes are hidden in Fluent by default because in typical flow problems (with Peclet number greater than 1), they're not good. To enable it regardless, you have to type in a command.
Code:
`/solve/set/expert no no no yes`
Quote:
 Originally Posted by arkie87 but as all those papers point out, False Diffusion" (as it is referred to in the literature) is a uniquely multidimensional phenomena. In 1-D advection equation, there is also a "false diffusion" in upwind scheme, which acts as a diffusive term, but this "false diffusion" is not usually what is meant by the term False Diffusion. If you go by truncation error alone, then 2nd order upwind should also not have any false diffusion, since there is no term proportional to the 2nd derivative of PHI.

ummmm,WHAT!? No. Volumes upon volumes of books have been written on this subject.

August 5, 2019, 11:37
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Raphael
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Quote:
 Originally Posted by LuckyTran Central schemes are hidden in Fluent by default because in typical flow problems (with Peclet number greater than 1), they're not good. To enable it regardless, you have to type in a command. Code: `/solve/set/expert no no no yes`
Thanks for the info. I thought it was Peclet greater than 2 (not 1)? And that is just stability with Guass-seidel solver. If you use deferred correction, there is no stability issue. To take care of oscillations, you can use flux limiter. Second order upwind (which is default) also needs flux limiter to prevent oscillations, so what is the difference?

Quote:
 Originally Posted by LuckyTran ummmm,WHAT!? No. Volumes upon volumes of books have been written on this subject.
Can you cite one? everything ive seen shows that there is no d2 PHI / dx^2 term. Only higher order terms... And if we are considering higher order derivatives, then central difference has higher order terms as well, no?

August 5, 2019, 11:45
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Filippo Maria Denaro
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Quote:
 Originally Posted by arkie87 i should also mention that in the literature, they distinguish between streamwise and cross-streamline diffusion. Streamwise false diffusion is what you are talking about (and is related to truncation error terms added from the scheme), whereas cross-streamline false diffusion is what is usually meant by "False Diffusion" as discussed in the literature. Regardless of the semantics, i havent seen anything that indicates that there should be zero cross-streamline False Diffusion for central difference scheme on uniform grid.

You are doing confusion in many issues...
First of all, let me address that terms like "diffusion" " dissipation", require a physical equation to be clearly addressed, it is not rigorous to talk of that only from the discretization of a derivative.

Then, the "false", "artificial", "numerical" diffusion are terms that you can find in literature.
First, be aware that an additional term in the form of a numerical diffusion (that is with a coefficient depending on the grid size) can be explicitly added in some formulations to the original discrete equation to stabilize the solution. This is the case of some "shock capturing scheme" where numerical viscosity is introduced in the Euler equations.

Said that, if you want to address the numerical viscosity "implicitly" introduced, not explicitly added, by the local truncation error you have to see the global time-space discretization and the so-called modified PDE. Doing that in 1D you will discover all you want from central and upwinding discretization. Determine the expression of the modified equation for each discretization starting from a linear convection PDE.

In a 2d (or multidimensional case) the effect of the flow alignement with respect to the grid direction is relevant in FD but can be alleviated in FV. However, you can find in literature the hystorical "tensor viscosity" representation to see the appearence of that. Again, you can analyse the modified PDE in 2D starting from the convection equation (use for the sake of simplicity the assumption u=v=constant and dx=dy).

Finally, you can analyse the modified wavenumber of a formula to see if it shows and immaginary part.

August 5, 2019, 12:01
#18
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Raphael
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Quote:
 Originally Posted by FMDenaro You are doing confusion in many issues... First of all, let me address that terms like "diffusion" " dissipation", require a physical equation to be clearly addressed, it is not rigorous to talk of that only from the discretization of a derivative. Then, the "false", "artificial", "numerical" diffusion are terms that you can find in literature.
agreed. but there is usually a clear distinction between false diffusion when applied to multidimensional case, and false diffusion in 1-D, where discretization error causes the solution to be equal to that of a lower peclet number (i.e. more diffusion).

Quote:
 Originally Posted by FMDenaro First, be aware that an additional term in the form of a numerical diffusion (that is with a coefficient depending on the grid size) can be explicitly added in some formulations to the original discrete equation to stabilize the solution. This is the case of some "shock capturing scheme" where numerical viscosity is introduced in the Euler equations.
Yes, but this is for stability, and has nothing to do with the false diffusion i am talking about.

Quote:
 Originally Posted by FMDenaro Said that, if you want to address the numerical viscosity "implicitly" introduced, not explicitly added, by the local truncation error you have to see the global time-space discretization and the so-called modified PDE. Doing that in 1D you will discover all you want from central and upwinding discretization. Determine the expression of the modified equation for each discretization starting from a linear convection PDE.
Yes, you mentioned this already...?

Quote:
 Originally Posted by FMDenaro In a 2d (or multidimensional case) the effect of the flow alignement with respect to the grid direction is relevant in FD but can be alleviated in FV.
Ok... this is the false diffusion i am talking about. I apologize if that wasnt clear...

Quote:
 Originally Posted by FMDenaro However, you can find in literature the hystorical "tensor viscosity" representation to see the appearence of that. Again, you can analyse the modified PDE in 2D starting from the convection equation (use for the sake of simplicity the assumption u=v=constant and dx=dy).
Ok, i havent seen that done in any textbook or paper so far. Maybe i will give it a try myself.

Quote:
 Originally Posted by FMDenaro Finally, you can analyse the modified wavenumber of a formula to see if it shows and immaginary part.
i'm not really interested in transient PDE, but thanks!

 August 5, 2019, 12:09 #19 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,764 Rep Power: 71

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