After further reading, my understanding is

1) starting from the original von Neumann theory (1947), considering what stated also by Trefethen (https://people.maths.ox.ac.uk/trefethen/4all.pdf), the analysis considers only the amplification factor of the numerical scheme, round-off or other errors is never explicitly considered in the original idea. The book of Anderson (4.5.1) also considers the round-off error not necessary. 2) we could argue why some textbooks (for example Hirsch) illustrated the theory on the basis of the round-off error evolution. 3) According to the statement of Trefethen, the stability could be analysed by means of the discretization error, too. However, the discrete equation for the evolution of such error is not homogenous, a fact that changes the evaluation of the analysis.

4) It seems that an analysis based on the Fourier transform of the modified equation could be an alternative and more accurate wave to evaluate the numerical instability.

5) Still not clear to me if and how the addition of the round-off error can be theoretically added.

No opinion?

Though I didn't understand the question completely, I put forward my suggestions. These are my personal statements may be wrong. I like to hear your opinion on it. It is not meant for criticizing Van Newmann stability analysis. Van Newmann is considered as the last legend in maths. Physicist got Hawkings but I think mathematicians didn't get any replacement for Van Newmann so far.

Later, I will include more justification in this report or my thesis. Since the paper dealt with combined space-time DRP analysis, I will post my opinion on it later.

Dear Arun Govind Neelan

the issue is that the original von Neumann analysis does not state at all any equation for an error (discretization or round-off). He analysed only the amplification factor of the discrete scheme. The assumption that the equation for an error is as same as the equation for the variable, appears later in the literature. And formally, this equation appears not correct.

Dear Filippo Maria Denaro ,

May I know your opinion regarding Von Newmann stability analysis? Once I able to find some time, I will make one document regarding DRP. It is quite hurting me that almost all the numerical analysis that I learnt is not suitable or imperfect. I blindly believed them and I took the wrong turn. This is not the first analysis I felt imperfect, positivity, monotonicity, Sweby diagram, dispersion relation preserving, stability analysis etc. seems imperfect for me. I'm totally confused, whether my understanding is wrong or numerical analysis is wrong!

I can say for sure that the original idea of the stability analsys was presented by von Neumann in terms of the amplification factor for the discrete equation governing the evolution of the variable, the equation for an error never being explicitly stated.

An example of literature where the round-off error was explicitly stated is in the Hirsch and in the Anderson textbooks. However, this latter also states that the introduction of an error in the stability analysis is not necessary.

The key should be in the fact that the physics of the problem should guarantee that the variable is bounded. That drives to a constraint on the amplification factor without introducing the error evolution.

My opinion is that if one want to study the stability in terms of an error equation, the formulation must be corrected as Ae= source, not as homogeneous equation

Dear Filippo,

The cows are coming home! As they say here locally! Stick to your analysis, as some of us have done this before and some researchers took a blinkered view. Of course, every scientist makes mistake(s)! There is always Beyond von Neumann analysis printed in JCP, vol. 226(2), 1211-1218 (2007) for assistance!

My opinion is that there is a gap from the hystorical works of famous numerical analysts (Courant, von Neumann, Richtmyer , Lax, Collatz, Morton) and what was interpreted in the CFD field. Even at present time, if you read the book of Threfethen the stability analsys is presented in terms of the amplification factor of the original scheme. In no way an error equation that has the same structure of the discrete scheme appears.

Conversely, in CFD textbooks you can find the equation for the round-off error or similar. But I want to highlight that Sec. 4.5.1 in the Anderson's book provides a "broader perspective". Even if the previous paragraphs illustrate the von Neumann theory starting from the debated error equation, there it is explicitly stated that:

"This might leave the incorrect impression that if we had a perfect computer with no round-off error, then there would be no instabilities. Such is not the case. The general concept of numerical stability is, in reality, based on the timewise behavior of the solution itself; it does not inherently depend on the behavior of the round-off error per se. "

Dear Tapan,

yes we already discussed about the analysis proposed in your papers. You followed a different way in the analysis, my only concern is about your definition of the discretization error as a continuous function instead of a finite dimensional vector. That has some severe implications.

Dear Filippo,

The implicit assumption in all computations is about continuity. Lots of opinion goes around characterizing computed shock as discontinuous. We all understand that piecewise continuous function (as in shock or fronts) are also continuous. If analytic solution (based on continuum model) is continuous and so is computed solution, then the error (which is the difference between the two) also must be continuous. I think I have cited the book by Shokin in this context, where the Gamma form finds the equivalent differential equation, corresponding to the difference equation for the error.

Have a nice holiday season.

Dear Tapan

thanks, we will keep in touch. I will continue to analyse the differences between the two approaches (continuos vs. discrete) for expressing a non homogeneous equation for the error evolution.

Have a good days

Dear Filippo,

When you continue, do keep the 2007 JCP short note in the perspective and write a critique on it, as it contains the differential equation for error dynamics. Our approach has been to trace error in numerical computing. If we do not know, what constitute error, how are we going reduce error and make it a scientific computing exercise?

Regards,

Tapan

Yes, I am already considering that paper. What I hope to understand is in terms of the spectral content of the error. Your continuos approach considers the infinite range of possible wavenumbers acting in the error spectrum, I am trying to isolate only the finite contribution up to the Nyquist frequencies but I still have the problem to consider the correct spectrum in the source term in the modified discrete equation.

I don't know if I would ever be able to do :-)

Best regards

Filippo

After struggling a lot, I deduced a discrete equation for the evolution of the round-off error, clearly different from what is classically addressed in literature. I post a page illustrating the idea.

Let me know your opinions.

Dear Filippo Maria Denaro ,

I like the way of treating error similar to analyzing the stability of iterative methods. I have few queries on it. From my understanding of Single value decomposition, if the system solution reduces over time (eg. heat equation) will have Eigenvalue smaller than one. Will it work for an unstable differential equation (solution increases over iteration/time)? I feel use of "other errors" rather than "round-off error" (error due to finite precision in computing) could be a better word choice. I think most of the time "round-off" error is not in our control and not a dangerous error in most of the present day computations. I will try this way of treating stability analysis on stable and unstable differential equations later.

I'm sorry that I was quite occupied so I was unable to find time for DRP schemes but I have started it. I need your help and Dr. Tapan K. Sengupta for that. I hope I will complete a draft within one week and I will communicate you through the mail.

Thank you.

Dear Arun Govind Neelan

the original von Neumann paper (1947) analysed exactly the heat equation. Even if the continuous differential equation is stable, the numerical discretization can or cannot be stable. An unstable continuous heat equation can be due only to the continuous addition of heat flux along the BCs that will increase indefinitevely the solution or due to the case of a negative heat coefficient that will increase the gradients.

The key is to understand first if the solution of the discrete equation, resolved without round-off error but with the discretization error acting on, is stable or not. That depends only on the eigenvalues of T. One sees that even without round-off errors the condition on the stability exists, based on the spectral radius. On the other hand, when the round-off error is acting is the analysis still correct or the conditions become different? From my analysis, it appears that the round-off error do not obey to the same discrete equation as conversely stated in the literature. Now the condition is on the rounded-off operator Tr

Dear Filippo,

I added a response yesterday. But for some reasons, I find it missing. The gist of the comment was reference to the following, as your analysis pertains to heat equation:

Article Error dynamics of diffusion equation: Effects of numerical d...

I hope you get to peruse it and it is not filtered!

Have a great day!

Dear Arun,

Please free to contact me whenever you need. But you have a great advisor in Prof. Nair, who is very conversant on DRP analysis. I have the privilege to work with him, during his dissertation.

Dear Filippo Maria Denaro

Sorry, I'm unable to completely understand the approach. If you post a report with a numerical test case, it will be useful for everybody. If you think, it has some scope, u can share it after publishing it. We lack good numerical analysis for non-linear equations and our governing equation is non-linear. I guess this approach may be extended to non-linear equations though Lyapunov stability analysis.

Dear Tapan

why the analysis should pertain to the heat equation? It is generally valid for any equation ..

Dear Arun

I am still writing a full report, I don’t know yet if it is enough valid To be submitted to some journal.

What do you mean exactly for adding a “numerical test”, using a certain numerical values for expressing the matrix Tr ?

Filippo Maria Denaro: Because the quote in your pdf of that original report talks of heat equation and talks of the physical solution to attenuate. We realized early enough that such attenuating model equations which do not propagate may not reveal the full range of issues of scientific computing. Still I provided the link of the JCP paper on heat equation to cross check your results with that publication. I would be interested to know your observations.

The most challenging equation is convection equation or convection-diffusion equation, both of which admit exact solution to calibrate your methods. The moment you include convection, you will have to talk about phase and dispersion errors. At this point, one can focus more on discretization or truncation error. Round-off error is too subtle to write down its characteristics. Also, a good error dynamic equation, along with global spectral analysis will give you clearly ideas of how round-off error will act. At least, we have been able to see it for convection equation.

But in the file I attached I considered indeed the pure convection equation... I just reported the original statement from the von Neumann’s paper...

Dear Tapan K. Sengupta

Dear Arun Govind Neelan

To better clarify the document I posted, the matrix analysis is general, the T and Tr operators can contain both advection and diffusion terms (and linear production term). At present my conclusions are:

1) if one would do the computation "by hand", that is at infinite numerical precision, the stability analysis drives to a constraint on the spectral radius of T. That is a numerical stability can arise also in a perfect arithmetic. This is the classical result of the von Neumann's analysis

2) In a real computation, one should see the action of the rounded-off operator Tr . It acts (by its power) in defining the evolution of the round-off error. Two distinct actions can be reasons of instability a) the action on the round-off error at the step zero and b) the shape of the initial condition. It appears that the classical constraint on the "perfect" operator T can be worstened in a real computation by the constraint on the real operator Tr .

The accuracy issue can be then analysed by the spectral analysis of the global error equation that is a sum of the discretization error and the round-off error equations.

Do not hesitate to correct me if you find some error.

Dear Filippo,

Let me think! I am yet to comprehend about the generic framework of your analysis, where the equation to be solved does not matter. Neither does it depend upon the discretization schemes used! With central spatial schemes, numerical amplification factor (G) is real for parabolic PDE. While for hyperbolic PDE/ wave equations, this G is complex. Thus for the latter, phase mismatch is the real deal! So I do not want to venture out and embark on a general framework! Also, each of our papers show that error evolution equation mimics the governing equation. Hence, I am not convinced by your analysis Filippo Maria Denaro . At least in the first glance. I need to think harder!

Tapan K. Sengupta the only assumption made in my analysis is the the starting PDE is linear. But in the matrix representation I introduced, you can in general include implicit/explicit scheme and any kind of central/upwind discretization for the derivatives of both advective and diffusive term. This way, in terms of the stability analysis you look at the behavior of the n-th power of the operators applied on the initial data. And that leads to a constraint on the spectral radius.

Successively, one can face the accuracy analysis by developing the amplification factor an the phase shift from the global error.

I could be wrong, for this reason I posted a brief page about my starting assumptions. Actually, I have a longer report on which I am still working on.

I shall wait to read your paper, when it is written and posted for all! Have a great Sunday!

Hey folks, it's "Neumann", not "Von Neumann". Von Neumann is a famous architect, Neumann is the physicist. (Trust me, I'm German. ;) )

Born as Neumann, "von" is after a law in 1900 integrated as part of the surname

Klaus-Michael Aye: It is amusing. He was born in Hungary and he changed his name to German version. I will not bother you with Wikipedia details, and you can edit it there, if you think that something needs changing.

Bringing back to light this discussion, for a further issue.

According to von Neumann, the numerical stability is analysed by considering that the original PDE does not produce new extrema and such constraint must be fulfilled by the numerical solution. This observation allows to analyze the discrete equation for the variable without invoking a specific form for the equation of the error evolution.

But what if the original PDE has a source term that allows for new extrema to be created? It seems that the starting von Neumann's hypothesis can no longer be used and somehow we should come back to the issue of expressing an equation for the error evolution. Isn't that?

Filippo Maria Denaro : This is the essence of our work since 2004, which shows the discrete equation for the error has three sets of such terms for 1D convection equation. The number of such terms will increase further for convection -diffusion equation for 1D convection equation. This is what you will call as the flaw of von Neumann analysis?

Tapan K. Sengupta

I mean flaw in the sense that the discrete equation is used to assess the conditions in which the amplification factor (for the discrete solution, not for the error) provides the stability condition. That is because it is assumed that the original PDE does not produce a solution with new extrema and we can check directlu the discrete solution. But what if new extrema are created by the real physics of the problem? It is still valid to require that the discrete solution is not amplified in a time step?

Filippo Maria Denaro

Indeed new terms appear for the equivalent differential equation for the error. Those terms ARE responsible for new extrema! This is another way of saying: Signal and error do not follow the same dynamics. The role of amplification factor is also explicitly visible for the error dynamics governing equation. For example, it is shown in our 2007 JCP short note that for 1D convection equation, we must have neutral stability to have no error arising from the numerical amplification term. So "yes" is the answer to both of your questions!