Hello, I'm a student in third cycle (applied mathematics), I work on the subject with finite volume method and I have a question that I don't find an answer.

An essential requirement for boundedness is that all coefficients of the discretised equations should have the same sign (usually all positive).

I don't know what is this phenomena and his mathematical signification.

please if you can help me!

I guess it is a property called positivity. If it is true. I can say something.

Let U(j)(n) be the numerical approximation at point j at time level n (in 1D or 2D or 3D). In general, you can write the update formula for U(j)(n) as

U(j)(n+1) = U(j)(n) + SUM_k [ Ck * U(k)(n) ]

where U(k)(n) are the numerical approximations at other grid points (usually only a few in 1D or several points in 2D) and Ck are the coefficients for these solution values. Note that SUM_K [Ck] = 0 for consistency (if all the U are the same, there shouldn't be any change for U(j)(n)). So, we can write

U(j)(n+1) - U(j)(n) = SUM_k [ Ck * { U(k)(n) - U(j)(n) } ]

Now suppose that U(j)(n) is the maximum. Then { U(k)(n) - U(j)(n) } are all negative. And if all Ck are positive, the change in U(j)(n) is negative, thus the maximum is decreasing. On the other hand, suppose U(j)(n) is the minimum. Then { U(k)(n) - U(j)(n) } are all positive, and therefore if all Ck are positive, the change in U(j)(n) is positive, thus the mimimum is increasing. The point is that the extreama do not grow. So, this gives stability.

Well, actually I have a question about this property, too. I still don't understand why this works for systems of equations where maximum/minimum growth is valid (consider velocity profile, for example, in a shock tube problem for 1D Euler equations. Initially zero velocity can become nonzero at laler time, thus maximum is increasing! ).

Your stability analysis is performed for the homogeneous case. In the general case, there are source terms so that maximum can increase. If equations are linear, the maximum of the error should decrease though...

franck

(1)Yes it is. And if there is a source term, yes, the maximum can increase. So, as I said, I don't undeerstand why this property is so popular and seems successful. For instance, Prof. Antony Jameson formulates his schemes based on positivity which he calls local extremum diminishing (LED). (2)I forgot to include the solution U(j) in the sum which leads to CFL condition in terms of positivity. (3)It can actually be applied to nonlinear problems. It is just that the scheme will no longer be linear if positivity has to be ensured. (4)By the way, it is not maximum of ERROR but the maximum of SOLUTION that decreases.

Nishikawa

I don't fully agree with your 1st and 4th statements: you can always imagine a source term (and a set of equations with unsteady solution) such that the maximum of the solution would increase in time. The concern of the numerical method is to find a numerical approximation to this solution. As a consequence, you are interest in reducing the maximum of the error. That is why the "maximum diminishing principle" applies to the error (homogeneous problem). Or am I wrong??

franck

I think Mr.Nishikawa maybe misunderstands the meaning of local extrema diminishing.The property is applied to the conservative variables rather than the primitive variables. For the computation of Sod's shock tube,the plot of the conservative variables,such as the density, actually satisfies the above property.Meanwhile I think the property is independent of whether the source term is considered.

Look in the book Computational methods for fluid dynamics by Ferziger and Peric - it is explained very well.

Hrv

Yes, density satisfies the property. But momentum does not. For example, consider the shock tube problem with initial data (density,velocity,pressure)=(1,0,1) on the left and (0.125,0,0.1) on the right. Because velocity is zero everywhere, momentum (density*velocity) is also zero everywhere intially. But the velocity does not remain zero everywhere once the diaphram is broken because the gas starts to move. A local nonzero maximum is created in the middle, and propagates to the right with a shock and left with an expansion.

Well, I thought I missed that point (conservative rathar than primitive) as you pointed out. But still, it doesn't seem to work.....

I understand what you're saying. Yes, in general, you want to find a numerical approximation to solutions whose maximum can increase. So, it makes sense to devise a numerical scheme that reduces the maxmum error, but does not make sense to devise a scheme that reduces the maximum value of the solution itself. But, as far as I know, the positivity property is discussed always in terms of solutions. For example, originally Harten introduced TVD (Total Variation Deminishing) in terms of the solution of Burgers equation. The word TVD is so popular that you can find it everywhere in CFD literature (for compressible flow simulations). Also, look at some Jameson's papers. LED is based on the solutions. It is actually an extension of TVD to higher dimensions (2D, 3D).

And yes, the "maximum diminishing principle" applies to the error if the problem is linear and homogeneous (I guess). But, the point is that you can apply positivity to nonlinear problems (such as Burgers equation) while von Neumann stability analysis is useless for nonlinear problems. So, looks like the positivity property is not related directly to errors in general... I think.

Let me make some comments on this discussion:

1. The maxima and minima in solutions to one-dimensional scalar conservation laws and the conservative variables in the one-dimensional Euler equations have special properties. These were first described by Boris and Book in 1973 and have been refined by various researchers over the years, e.g., Harten. Basically, maxima do not increase in time, minima do not decrease, and no new maxima or minima are created. Many conditions have been proposed to ensure that numerical schemes inherit these properties, in part or in whole, e.g., ENO, TVD, and TVB. These are best understood as nonlinear stability conditions.

2. The properties of maxima and minima in conservative variables of the 1D Euler equations do not hold at discontinuities, namely, shocks and reflecting solid boundaries. Essentially, at discontinuities, one conservative variable can take from another, allowing maxima to increase in one conservative variable at the expense of another. However, one can devise sum totals over all conservative variables that will hold up, even at discontinuities.

3. Viscous terms, chemical source terms, etc., affect maxima and minima. They no longer have the same properties as the raw Euler equations. One approach is to discretize the Euler terms using approaches based on preserving the properties of maxima and minima in the Euler equations, and doing something else with the other terms in the equation.

4. The TVD property means that a signed sum of extrema does not increase. In other words, the sum of all maxima minus the sum of all minima does not increase. As far as I know this result first appeared in Laney and Caughey, "Extremum Control II: Semidiscrete Approximations to Conservation Laws," AIAA Paper 91-0632.

5. TVD works well enough in 1D, although most "TVD" methods actually enforce much stronger conditions than TVD itself. These include the positivity conditions that led off this discussion. Harten can be given most of the credit for bringing positivity conditions to nonlinear approximations. Before that, they had mainly been used in linear approximations.

6. TVD does NOT ensure that maxima do not increase, minima do not decrease, and that no new maxima and minima are created. Strictly speaking, it resricts only the sum total of maxima and minima.

7. TVD does not work in multidimensions, where it limits accuracy to first-order. That explains the move towards focusing on individual extrema in multidimensions, rather than sums of extrema. As far as I know, the first to suggest this is focus is Laney and Caughey, op. cit.

8. Jameson, with terms such as LED, has developed the ideas found in Laney and Caughey, op. cit. If you read his early work in this subject area, you will see that he cites Laney and Caughey, op. cit. (This is not to say that Jameson has not contributed immensely to the understanding of the pertinent issues.)

9. Even though this is a long response for a bulletin board, there is much more to be said on these issues. I am always surprised that people expect this kind of detailed technical discussion on a bulletin board. It generally turns into a case of the blind leading the blind. If you want to understand these issues, you need to consult the literature. You will find a thorough discussion of these issues in Laney, "Computational Gasdynamics," Cambridge University Press, 1998. See especially Chapter 16. A web site is http://capella.colorado.edu/~laney.

Yes, I am blowing my own horn here somewhat. However, this discussion has hit squarely on my area of expertise and contribution, so I hope that I can be forgiven. Like most researchers, I usually ignore these types of discussions, believing that people should do their homework before asking for clarification, but this one I found just too provactive.

Bert Laney

This was most informative, and I, for one, don't care about "blowing my own horn" as there is clearly worthwile information being presented. I am going to purchase the book referred to! Thanks, Dr. Laney

Your message is very useful, Dr. Laney; a good introduction to the homework.

Hello ;

I have some problems in writing the discretize governing equtions for cavity flow (laminar). Thanks a lot to every one who help me.