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Central Differencing Scheme and False Diffusion 

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August 5, 2019, 08:02 
Central Differencing Scheme and False Diffusion

#1 
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Raphael
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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 gaussseidal/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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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 

#3  
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(2) it is not available in Fluent for laminar problems? 

August 5, 2019, 08:53 

#4 
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August 5, 2019, 09:07 

#5  
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Filippo Maria Denaro
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Quote:
 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 

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August 5, 2019, 09:17 

#7  
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Raphael
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Quote:
(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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Are you saying that a linear advection equation needs to be made nonlinear somehow (e.g. using a flux limiter) in order to allow for a nonoscillatory solution when solving using central difference (or any higher order scheme)? Thanks! 

August 5, 2019, 09:22 

#9  
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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 nonlinear the numerical scheme even if the PDE is linear. 

August 5, 2019, 09:27 

#10  
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Filippo Maria Denaro
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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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(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 nonzero 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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1no, 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 

#13  
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but as all those papers point out, False Diffusion" (as it is referred to in the literature) is a uniquely multidimensional phenomena. In 1D 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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Regardless of the semantics, i havent seen anything that indicates that there should be zero crossstreamline False Diffusion for central difference scheme on uniform grid. 

August 5, 2019, 11:08 

#15  
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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:
ummmm,WHAT!? No. Volumes upon volumes of books have been written on this subject. 

August 5, 2019, 11:37 

#16  
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Raphael
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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 

#17  
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Filippo Maria Denaro
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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 timespace discretization and the socalled 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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i'm not really interested in transient PDE, but thanks! 

August 5, 2019, 12:09 

#19 
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Filippo Maria Denaro
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central difference, false diffusion 
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