The answer to a question about the numerical algorithms for resolving the issue of differentiability of a function is typically provided by the textbooks on experimental mathematics.

I recommend in particular: Chapter 5: “Exploring Strange Functions on the Computer” in the book: “Experimental Mathematic in Action”.

For the review please see

https://www.maa.org/press/maa-reviews/experimental-mathematics-in-action

You can also get a copy of the text in a form of a preprint from

https://carma.newcastle.edu.au/resources/jon/Preprints/Books/EMA/ema.pdf

Judging by the quote placed in the beginning of Chapter 5, the issue of investigation of the “strange functions” was equally challenging i 1850s as it is 170 years later:

“It appears to me that the Metaphysics of Weierstrass’s function

still hides many riddles and I cannot help thinking that enter-

ing deeper into the matter will finally lead us to a limit of our

intellect, similar to the bound drawn by the concepts of force

and matter in Mechanics. These functions seem to me, to say

it briefly, to impose separations, not, like the rational numbers”

(Paul du Bois-Reymond, [129], 1875)

The situation described in your question is even more complicated because the function is represented only by a few values on a rectangular grid and it is additionally assumed that the function is not differentiable at a certain point. In this situation I can suggest to use the techniques employed in the theory of generalized functions (distributions).

For a very practical example you can consult a blog: “How to differentiate a non-differentiable function”:

https://www.johndcook.com/blog/2009/10/25/how-to-differentiate-a-non-differentiable-function/

In order to answer your question completely I would like to know what is the equation, boundary conditions and the numerical scheme used to obtain a set of the grid point values mentioned in the question.

Is the function continuous at all points? If it is discontinuous, then you may approach the discontinuity from left and right to see if the limits approach one another.

Numerically speaking, the process is equivalent, only, you must set up your algorithm to perform the derivatives from left and right directions.

Yes, the function ( or the field I may put it more appropriately ) is continuous.

It's like solving standard 2D heat transfer like PDE ( for some scalar ) and the scalar along some direction attains the profile as mentioned.

I get your point to take derivatives from left and right directions. But in 2D field , as mentioned above, how will defined left and right.

Say the variable is X at grid point (i,j) in uniform cartesian grid and we want to check along the normal direction which may be , lets say 60 deg inclined with x-axis. While getting normal vector is straight forward, how to check which is left and right in such scenario ?

Thanks

Deewakar Sharma

I think your question is somehow hill-posed. A numerical derivatives is nothing but an approximation of the continuous derivative. If this latter does not exist at some point, it is wrong to assume a discretization of this derivative. FD-based derivatives assume by definition that the function is regular and differentiable.

In practice, you could see that, given a singularity at Xs, you can compute left and right asymmetric discrete derivatives in Xs and they do not converge to the same value while refining the grid.

If you suspect that your multidimensional function is not differentiable, then try to parametrize the directions on your grid, and see for which values of the parameter(s) the differentiability definition does not work.

You seem to assume continuity. However, both a function does not have to be continuous in order to be differentiable, nor vive-versa.

@Deewakar sharma

Check out page 4-8, title: "a135900," of the attached document. I would encourage you to employ methods from conformal mapping and complex analysis to your problem. The other paper, "fluids2," is pertinent as well.

It is only by convention the righthand rule is used in physics, also, for 2D domains, counterclockwise direction is the conventional direction to take. So, calculate normal vector, assign the lefthand side the convention, rather, counterclockwise direction. It gives a rule to apply to the entire domain. That's all.

Now, on the aside, you shouldn't experience any discontinuities in your domain, unless, you are calculating from a multi-regional domain. In that case, differring properties, like conductivity, can cause boundary problems; but, you should, or, the algorithm should be taking care of such situations and assuring the boundaries match.

Would need to hear more from you to go further...

Luisiana Cundin thanks for your files. I will have a look and update about as soon as possible.

The answer to a question about the numerical algorithms for resolving the issue of differentiability of a function is typically provided by the textbooks on experimental mathematics.

I recommend in particular: Chapter 5: “Exploring Strange Functions on the Computer” in the book: “Experimental Mathematic in Action”.

For the review please see

https://www.maa.org/press/maa-reviews/experimental-mathematics-in-action

You can also get a copy of the text in a form of a preprint from

https://carma.newcastle.edu.au/resources/jon/Preprints/Books/EMA/ema.pdf

Judging by the quote placed in the beginning of Chapter 5, the issue of investigation of the “strange functions” was equally challenging i 1850s as it is 170 years later:

“It appears to me that the Metaphysics of Weierstrass’s function

still hides many riddles and I cannot help thinking that enter-

ing deeper into the matter will finally lead us to a limit of our

intellect, similar to the bound drawn by the concepts of force

and matter in Mechanics. These functions seem to me, to say

it briefly, to impose separations, not, like the rational numbers”

(Paul du Bois-Reymond, [129], 1875)

The situation described in your question is even more complicated because the function is represented only by a few values on a rectangular grid and it is additionally assumed that the function is not differentiable at a certain point. In this situation I can suggest to use the techniques employed in the theory of generalized functions (distributions).

For a very practical example you can consult a blog: “How to differentiate a non-differentiable function”:

https://www.johndcook.com/blog/2009/10/25/how-to-differentiate-a-non-differentiable-function/

In order to answer your question completely I would like to know what is the equation, boundary conditions and the numerical scheme used to obtain a set of the grid point values mentioned in the question.

Janusz Pudykiewicz

Thanks for such a detailed suggestion. I will surely have a look. Just to answer last part of your question,

I am using finite volume, central difference for the scalar field I am trying solve. To give you can idea, here is the picture I get in post processing along on line.

As you can see, there is peak or point of non-differentiable, though numerically if we do, we may get some answer. However, what I want is to detect at this node.

Thanks

Deewakar Sharma

now you addressed about the finite-volume formulation but that has a deep implication on your original question. The FV method is based on the discretization of the integral (weak) form of the equations. This way you can manage non-regular functions (i.e., discontinuity) but you have to be aware about the meaning of the resulting computed solution. Indeed that you get is no longer the pointwise solution but a regularized smooth (weak) solution.

Again, we could debate a lot about the requirements of any type of discretization for either finite-difference or finite-volume formulations.

If you assume that your weak solution is a regularized continuous function, the you can estimate the max norm of the gradient of the function to check for the presence of a spike in the solution and its direction.

Filippo Maria Denaro

Thanks for your reply. Man you elaborate what you mean by checking presence of spike using max norm of gradient. If I get right, then evaluating norm of gradient is going to give component of gradient along normal direction. And ideally, at the spike or lets say where I encounter sudden change in slope, I should expect something weird ( theoretically 2 slopes ) but how does max norm explain this. I did not get that.

Do you mean that " weird " part will be relatively quite high as compared to all?

Deewakar Sharma

first of all, when you compute the gradient of a scalar function you get a vector field, that is you have magnitude and direction of the vector at any discrete point.

You can use that to adopt the max norm on the discrete set of the gradient components. As you compute a FV-based solution this latter appears "smeared" and regular but will show the high gradients. This is a signal where the pointwise function is discontinuous. Your previous plot already addressed what I mean.

If you switch to seeking maximas, then numerically speaking, this is a huge problem with no real solution. There exists algorithms to find local maximas:

https://www.mathworks.com/help/signal/ref/findpeaks.html

https://www.datasciencecentral.com/profiles/blogs/optimization-techniques-finding-maxima-and-minima

I think @Filippo says it right...the FV gives weak solution, therefore, discontinuities are not in your set, but you obviously do have local maximas.