Artificial numerical diffusion
 

Hello Friends,

In recent days, I am concerned with fluid flow in high Reynols number. In these flows, it is now well known that central difference may produce unphysical oscilations in particular regions of the flow. In adition, first order upwind overcomes this instabilities, but it introduces numerical viscosity. Thus, I would like to known the source of this artifical numerical viscosity. Please, give me examples and papers, if it was possible.

Thanks.

Re: Artificial numerical diffusion
 

(1). This has been discussed here before. If you have time, you may be able to find some upwind related messages. (2). There is a better way to look for the answer. In the forum/resources/documents/books online/ here, David Creech has compiled a complete list of CFD related books. P.J. Roache's 1972 book entitled "Computational Fluid Dynamics" was the first of this kind published. I think, it was based on his PhD dissertation, so, he knows what he is doing. That was the book I bought in early 70's when it was first published. (3). If you can't find it, the 1998 edition entitled "Fundamentals of Computational Fluid Dynamics" should be available by the same author and the same publisher. I don't have the new edition of the book. But I think, you can trust a CFD expert with 30 years of experience like Roache. If you have money to buy more books, you can use David Creech's list as a guide.

Re: Artificial numerical diffusion
 

Hi, Valdemir,

John is quite right, you should read those books. But today is Sunday, some people go to church, and science is my religion, so ... :)

Consider 1D convection-diffusion equation, with a=const>0 and k=[viscosity]=const>0:

(1) a du/dx = k d^2u/dx^2

Upstream-differencing approximation on a uniform grid is:

(2) a (u(i)-u(i-1))/h = (u(i-1)-2 u(i)+u(i+1))/h**2

with x(i+1)=x(i)+h implied.

Clearly, solution of (2) is not a solution of (1). Ask yourself, what is the differential equation whose solution coincides with a solution of (2) at points x=x(i)?

Assume u(i)=U(x(i)). Expand U in a Taylor series about x=x(i):

(3) U(x)=U(x_i)-dU/dx*(x-x(i))+d^2U/dx^2*(x-x(i))**2/2+(...)*(x-x(i))**3+...

Substitute (3) in (2):

(4) a dU/dx = (k+a*h/2)*d^2U/dx^2 +(...)*h**2+...

Neglect everything multiplied by at least h in (4). Then (4) coincides with (1). We say that (2) is a first-order approximation of (1). If, however, you do not think h is small enough to be neglected, you can neglect terms with at least h**2. This gives

(5) a dU/dx = (k+a*h/2)*d^2U/dx^2

This is called a differential approximation to (2).

You can see, that if O(h) terms are not neglected then (2) again approximates the convection-diffusion equation but with greater viscosity: k_total=k+k_numerical, k_numerical=a*h/2. Repeat this on paper yourself.

Kghm, you surely know this, but, just for beginners, it remains to note that all this does not ensure yet that by solving (2) one gets approximate solution of (1), (2) approximates (1) all right, but one also needs to prove that (2) is stable, and only then, by Central Theorem of Numerical Analysis, which is (approximation+stability => convergence) one can conclude that the solution of (2) tends to the solution of (1) as h->0. To learn more about this one certainly should read books.

Have a good weekend :).

Sergei

Re: Artificial numerical diffusion
 

When talking about accuracy of numerical solutions, cfd-people usually imply only false (or, artificial) diffusion. Sure, that IS very important part of the numerical error. But, guys, keep in mind that there is also numerical dispersion which may be quite large to kill your numerical solution...

Re: Artificial numerical diffusion
 

Hi Dear Friends: John, Sergei and COBOK. Thanks for all. I have some questions: first, is the equation (5), placed by Sergei, the modified equation of (1) at which we, in fact, are solving on a computer? second, Can I extrapolate this numerical diffusion for 3D Navier-Stokes equation? and finaly, for high Reynolds number the fluid will not be modified?

Have a good week. Valdemir.

Re: Artificial numerical diffusion
 

a discussion of artificial dissipation/dispersion (whether implicit or applied) can be found in any intro text on cfd you can try any of the list of books on culbert laney's site www.capella.colorado.edu/~laney/otherbooks.htm many of these should be in any university library.

Re: Artificial numerical diffusion
 

Dear Valdemir,

You see, we can answer your further questions but new questions will appear and so on. Note that people here recommend you books, not papers, and there is no way out of this. It is a question of volume. One cannot expect a chapter of a book being reproduced here. So, go ahead with books.

In a computer we solve (2). Both (1) and (5) describe the solution of (2) approximately, with (5) giving better approximation.

The reasoning can be extended to 3D case.

Good luck.

Sergei

Re: Artificial numerical diffusion
 

Take a look at "Heuristic Stability Theory" by C. W. Hirt. It's a paper in the Journal of Computational Physics, somewhere in the 1960's.

The paper has an unusual clarity even today, and it's possible to see how the results extend to 3D.

Good Luck!

Re: Artificial numerical diffusion
 

Thanks Jim. You understood me.

Good Luck for you too.