Hello Shashank,

  Attached is a book, Patanker's: Numerical Heat Transfer and Fluid Flow, where on page 108 you will find information on the false diffusion. Additionally, I've added a paper written by M.K. Patel, N.C. Markatos, M. Cross, "Method of reducing false-diffusion errors in convection - diffusion problems", which gives greater detail for the derivation of the false diffusion. Also, DTIC technical paper on deriving finite difference formula, should have an explanation for the false diffusion.

Hello,

the numerical diffusion can be deduced from the analysis of the Local Truncation Error (LTE) of the scheme (form the convective term). After the time and space discretization is chosen (typically the numerical diffusion is due to the use of upwinding), you use a Taylor expansion and see the leading term in the LTE. Numerical diffusion does exist also in the 1D problem I suggest starting from 1D and then going to the 2D case.

Many CFD textbook illustrate this analysis.

hi Luisiana and Flippo

Thanks for your answers. I believe I was not very clear in my question.

My specific doubt is : How to prove numerical diffusion is a multidimensional phenomenon and it does not exist in 1D cases ( although truncation error analysis proves otherwise). The same is mentioned in the Patankar book and the paper as well.

The paper says"

This can only be explained by the diffusion coefficient derivation which will make FD to be zero in 1D case since theeta is zero.

But when I try to do a similar analysis for 2D cases, I get artificial viscosity coefficient as

0.5*U*del(x)+0.5*V*del(y).

Attached is my derivation.

Thanks

well, I totally disagree with the statement that the numerical diffusion exists only in the multidimensional case... the multidimensional case produces a matrix of coefficients for the numerical diffusion while the 1D produces a one-coefficient term.

Hello Shashank,

For the analysis of 1-D case please consult equation (8) in the paper:

http://twister.caps.ou.edu/CFD2015/SmolarkiewiczMWR1983.pdf

The derivation is very instructive and it provides an explicit formula for the numerical diffusion coefficient in 1-D case.

Janusz

Many text books have derived (Taylor's series) and discussed about the approximate coefficient of numerical diffusion. But in your question "prove". Then please refer the text books that is having the approximate coefficient of numerical diffusion.

@R.C. Mehta : Thank you for your answer.

My doubt is the discussion over books and articles does not match with what is being explained in Patankar and the paper.

Patankar book states that numerical diffusion is zero when the grid is aligned or orthogonal to the direction of flow.

It means that false diffusion has to be zero in 1D cases.

This can be justified by the formula given in the paper and book but I am not able to derive the formula.

When I do finite difference formulation and collect the second order differential terms in truncation error I get simply

Numerical Diffusion Coefficient as u*delx/2+ v*dely/2

Thanks

It's a long time ago since I worked with the stability to solve N-S equations. When problems with the stability occured I used upwind method or even increased the viscocity. I.e. false diffusion. A better way was to make a finer mesh

Interesting papers may be: "The solution of the Three-dimensional Navier-Stokes Equations using a new finite difference approach", D.F.Roscue, University of Sheffield, England, Int. Journal for numerical methods in engineering, Vol. 10,1299 1308 (1976)

In this paper the author used a method which balance between central differences and upwind differences by help of ex-functions. When stability problems occurred you just multiplied upper and lower terms  with e-x ,

Hello Shashank,

The effect of numerical diffusion is observed in both 1-D and multidimensional cases. When deriving an explicit formula for the corresponding diffusion coefficient we apply the method described in the paper mentioned in my first letter (please see the comments before eq. (7)). In summary, we first write our scheme in an algebraic form and then the terms $\phi^{n+1}_i$, $\phi^n_{i+1}$, $\phi^n_{i-1}$ and so on, are expanded in a second order Taylor series around the point $t^n, x_i$. After this step, we obtain an explicit form of the coefficient appearing in front of the second derivative with respect to the space variable $x$. In the case of a multidimensional scheme the equivalent procedure is based on the expansion using the Taylor series for the function $\phi(t,x_1,x_2,...,x_n)$. This procedure will quickly lead to rather complex algebraic expressions and the use of a symbolic algebra systems (for example Mathematica) is very helpful.

In 2-D case you will have a diffusion tensor instead of the diffusion coefficient because the scheme will have a cross derivative terms in addition to the standard diagonal terms with the second derivatives with respect to $x_1# and $x_2$.

Please note that you have two omissions in your derivation:

1. the expansion in Taylor series is substituted to the continuous equation instead to the corresponding discrete approximation

2. the expansions are performed only around $x$ instead around $(t,x)$.

I hope that these comments are useful.

All the best,

Janusz

Dear Shashank Mishra,

Patankar was a good texbook but had some weakness...I suggest a reading of modern CFD and numerical analysis textbooks. Furthermore, the  denomination of "artificial viscosity" is due to the large use of the first order upwind (FTUS) method in the first CFD simulation done many years ago. The local truncation error analysis can be done both in the physical space and in the wavenumber space, leading to the modified wavenumber analysis that I suggest to looking for. You will find that, for the 1D upwind  schemes of any order, the modified wavenumber has also an imaginary part, giving reason of the presence of artificial diffusion (dissipation for higher order terms).

You could be interested in an automated procedure for determining the expression of the local truncation error and look for the presence of numerical diffusion/dissipation

https://www.researchgate.net/publication/233758608_Using_symbolic_computation_software_packages_in_production_of_multidimensional_finite_volume-based_large_eddy_simulation_codes

Article Using symbolic computation software packages in production o...