Hello Aditya!

Both equations tell the same tale: the conservation of momentum. One is formulated for  an infinitesimal fluid particle (the differential form) whilst the other is applied to a region in space, the control volume. These two are linked by the Gauss theorem, and can be derived via almost ten different ways. Both originate from the same first principles (the Newton's 2nd law of motion). Mathematically, they are equivalent in the infinitesimal limit.

For numerical solutions, the differential form is used together with difference approximations (FDM, finite difference method). Integral form is used with the finite volume method, FVM. These are equivalent in uniform grids. The differential form does not have a solution in the classical sense in presence of discontinuities (eg. compressible flows with shocks), hence, one uses the weak form of the integral equations. A physically unique solution is sought using an entropy condition.

The conservative property has a numerical aspect, concerning the overall conservation. This is usually important in an approximate solution; FVM are better in this respect. For example, a conservative momentum flux is \nabla . \rho u u, while u . \nabla \rho u is not. In differential form these are again equivalent but one can never can shrink the grid spacing infinitely to ensure the solution of the differential equations is that of the integral equations applied to an infinitesimally small control volume.

Thank you for the explanation ... Ville Viitanen and Bernhard Semlitsch ..

I am more used to finite element formulation of NS equations. In this approach ... one uses the weak form of the differential equations, I am interested to know how different is this weak form from the weak form Ville Viitanen mentioned, that is weak form obtained from the integral form of the NS equations.

Because in the finite element approach an integral is used for approximation can it be said that, even though differential form of the NS equations are used,  the finite element approach can be used when there are discontinuities (shock) in the flow ?? If not what would be the difference.

Also is the convective term that is "\nabla . \rho u u" or "u \nabla \rho u"  only responsible for the non-conservativeness of the equations ... ? If yes .. it will be interesting to know how ...

The differential form mean to divide the control volume into a large number of small elements and apply the continuity, momentum, energy ...etc as in CFD software.

The integral form consider the control volume as one element and apply the  mass, momentum and energy equations. 

FEM utilizes the concept of weak solutions, where from the whole space of test functions one choose a particular subspace of usually piecewise constant or linear functions in the simplest cases. But it is the same concept particularised.

This issue is quite fundamental both mathematically and numerically...

The integral form is direct consequence of the Reynolds transport theorem, the differential form is derived from the integral one provided that some regolarity conditions exist and the differential operator applies.

Formally, the integral form (it is demonstrable that it is a weak form) has a lack in the fact that produces a solution that is not unique. Mathematical but non physical solutions are possible. This is typical of inviscid compressible flows. Non regular solution can be treated but are not unique if you do not provide some additional condition.

Numerically, the integral form leads to the Finite Volume discretization (or Finite Elemento) while the differential form leads to the Finite Difference discretization (also Spectrral Methods adopts the differential form).

FV are by construction conservative, FD can or not.. The conservative property ensures the correct waves propagation.

Last, the use of the integral form can be linked to the adoption of volume average in LES

Some useful books:

1) Ferziger & Peric

2) Leveque

Totally endorsing Ville Viitanen..here are few more comments in terms of  physical explanation:

1. Governing equations are generally derived in differential form as they are derived by flux balance over infinitesimally small control volume...

2. In application to certain problems, the integral forms are convenient, viz., in fvm, where the differential equations obtained for infinitesimally small control volume are integrated over finite control volume

3. Essentially both are same, result of Newton's second law or force balance over control volume 

4. To solve a particular problem, depending upon domain and method used, appropriate governing equation either in differential or  integral form is selected

I would highlight that the governing equation are derived in the integral form, not in the differential. That is stated by the Reynolds transport theorem.

Furthermore, provided that the regularity is ensured, the differential form can be written in terms of divergence (conservative) or quasi-linear form.

Last, for viscous flows the integral form still require regularity for the diffusive flux, there is not waek form in such case

Dear Aditya,

 Actually, in the frame of continuum mechanics, the integral forms of the equations of conservation of mass, linear momentum and energy are firstly derived. Of course, there is no trick  : if you consider a newtonian fluid for instance, the angular momentum conservation is taken into account by requiring that the stress tensor is symmetric. For this purpose (i.e. derive the integral forms), you can adopt lagrangian point of vue by means of the Reynolds transport theorem on the one hand. And, on the other one hand, you can either adopt the eulerian point of vue, considering a control volume V. The derivation of each integral form and the inference of the differential forms of each equations of conservation are really straightforward  (p57, 2.19 e.g. ).

You are right about "conservative", even the use of "conservation" of linear momentum is puzzling at first ; it is not conserved unless the resultant force is zero, isn't it ? In conservative forms, rho, rho u and rho e are unknowns. They are used for compressible flows when neither rho is constant nor div u is zero). The convective part of the total derivative is of the form : "divergence of something". And, at that point, you understand why these forms are called "conservative".

Regarding your second question, I would add a very simple example of an « every-day-use » of the (euler) integral form of linear momentum which is used to carry out an estimation of the resultant force in a Venturi or in a curved pipe.

Hoping that it could help.

conservative forms are related to divergence property .partial differential equation derived from fluid element fixed in space is called conservative form and fluid element moving in space is called non-conservative form

be careful that FD and FV do not produce (apart few simple cases) the same algebric system, even on uniform cartesian grids

If you have a converged solution for both methods (and use the equations you solve, the discretization, b.c.'s etc are the same) then actually there is no real difference between the two. (When a solution can actually be called converged is another topic, but say the residuals are up to machine accuracy)

The main difference I would say is in the usage: finite difference is easier to implement (and easier to explain), you can use higher order discretization schemes, you can more easily use fancy stuff like multi-grid methods, but you need a structured grid, which can be a limitation. Actually, the finite difference method allows for loss of mass which is not possible for a finite volume method because of its formulation. But that is only when the solution is not well converged.

Finite volume has the benefit that for your grid you can use any kind of volume: tetrahedral, hexahedral, polyhedral etc. Some codes can also deal with hanging nodes or Cartesian grids. This gives a lot of freedom in the geometry that you want to use and in principle you can more easily make local grid refinements which makes your grid and computation more efficient. The disadvantage is that it can never be better than second order (to my best knowledge).

The term 'conservation form' and 'non-conservation form' could be called a misnomer since both forms are derived from the conservation laws and it has nothing to do with that (I know mathematicians have a good reason for this name). It is just a name for a mathematical property.

Well, I do not totally agree ... FD comes from local differential formulation, FV comes from the integral formulation, the different mathematical nature being relevant in understanding the real type of solution. Think about the term d/dt Int [V] f dV, that produces nothing else but the time variation of the average quantity f_bar. Mathematically, only if |V|->0 then f_bar-> f locally. That means that using FV discretization on a practical grid of small but finite measure, the solution one gets is always a volume-average that (on centered cell) is a second order approximation of f, even if you use high-order (more than second order) reconstruction for the fluxes.

I don't see where we seem to disagree. My point was also that using FV you cannot get higher than second order (also because of the use of unstructured grids*). And your point that only if |V| -> 0 then f_bar -> f is true for both FV and FD...

*) even structured grids are usually treated the same as unstructured grids in FV.

Dear Arjan,

there is a relevant mathematical difference in using either the strong or the weak form of the equations. You cannot use the FD discretization of the strong form when solutions are not regular. As counterpart of the continuous formulation, even the discrete formulation can produce wide difference when using FD or FV. Such differences are relevant as going to |V|->0 is only a formal consistence requirement. Usually we have a finite measure and both accuracy and stability can be strongly dependent. Just an example, the FD discretization of the quasi-linear form does not capture the correct waves progagation and does not ensure conservation.

Regards

Conservation laws are always first written in the integral form. Should the solution of a particular problem be "sufficiently smooth", this integral form is equivalent to the differential form of the same. Message: integral forms of conservation laws admit a wider solution space (with lesser continuity constraints) than the differential forms of the same.

I suggest you read Prof. Enzo Tonti's book: The Mathematical Structure of Classical and Relativistic Physics, Birkhäuser (Springer) (2013). His most recent papers advocate directly solving the integral forms of the conservation laws. I believe your doubts will be clarified.

Conservative forms are those forms whose discrete counterparts inherit the conservative structure. In other words, the discrete equations (say obtained by FVM) express balance of discrete fluxes. This conservative structure at the discrete level is a desirable property and essential for certain applications, e.g. compressible flows which develop shocks. The discrete equations of non-conservative forms do not express balance of discrete fluxes. In compressible flows which develop shocks, the solution of non-conservative forms might predict incorrect location of shocks.

I suggest you read Prof. Culbert B. Laney's book: Compuatational Gasdynamics. Book DOI: http://dx.doi.org/10.1017/CBO9780511605604

I'm a CFD guy, so I have a bias.

In short, the integral form comes from a macro balance, as shown in many Fluid Mechanics books, and is used in the derivation of Finite Volume Methods when the macro volume reduces to a small elemental volume, while the differential form means a micro balance, and is used to get the Finite Difference discretization, and to begin the Finite Element approach.

Applying the Divergence Theorem, and the demand that the macro balance will be valid for any field, leads to the differential form.

All the forms are conservative, in the continuum, only when discretized they split in two ways: one identifying a volume over which conservation is applied, and the other being applied on a point, over which balances are meaningless. It's more a jargon then a precise definition.

hth

Just to add some issue....

- FV is a particular FE formulation

- Smooth initial condition can still prorduce high gradients and conservative vs non-conservative discretizations show a high impact on the correct wave velocity representation. This is relevant in turbulence, e.g. using LES.

- a simple example of strong difference in the algebric system coming from second order discretization in 2D :

FD: d/dx (u^2) -> (u^2(i+1,j)-u^2(i-1,j))/2dx

FV: Int [-dy/2:+dy/2]  (u^2(xR,y) -u^2(xL,y) dy -> Use the trapezoidal rule and see the difference...

by definition, integrals allow a number of breaks in the medium while differentials are determined only for continuum. Thus, if you have domain where the ruptures are available, the integral equations may be used but differential not.

The Differential form is applicable at a point while an Integral form is applicable to an extened region.

Deepak Kumar If this anwser was not written, I would have done it :) Cheers

Quite old post, I would add this recent paper that highiglights further topics

Simply, if you are interested in qualitative nature of flow you may use integral form, otherwise for quantitative and subscale molecular behavior of flow you have to use differential form of NS equations.

Integral describes the behavior of the flow in boundaries like interface, while differential explained the behavior in small scales like eddies, turbulence, and momentum transport.

Considering CFD, the Differential forms apply fundemental laws such as conservation of mass and momentum to infinitesimal fluid particles, in contrast, the integral forms apply those laws to finite volume in flow domain. Refer to White book for better understanding

the integral form is more accurate than differential form

Despite this thread is old and some good answers were provided, it seems that the issue appears still not clear.

The only physically relevant formulation is the integral one, according to the transport theorem it has a specific physical meaning and conservation property on a material volume of finite size.

The passage to the point-wise differential formulation is a mathematical formulation, provided that the functions are regular. Furthermore, the "point-wise" formulation makes sense only by assuming a small but finite volume around the general position (continuum hypothesis).

Thanks guys, Filippo Maria Denaro especially. Is there any type of problem where one method is favoured over the other? is it fair to say the FDM is less computationally expensive?

All flow problems where discontinuity appears can be formulated only in weak form, that is, equivalently, in an integral form.

thanks Filippo Maria Denaro I thought people just said above that the differential form was in weak form?

Not at all!

Brett Dewar

I suggest a look to the Leveque textbook, you will find a proper section about the weak form.

What you mentioned about the differential form has to be better clarified like the differential form applied to the local averaged variable, not to the pointwise variable.

The differential and integral forms have their on advantage for various numerical methods. For example, the differential form is usually used for the finite difference method and the integral form is used for finite volume and finite element methods.

The integral equations contain additional scales of length and time, which appear in the limits of the integrals. Because of the nonlinearity of the equations, those scales will appear in the solutions of the equations. More particularly, they appear as the coefficients of higher-order derivatives that represent dissipation. In the numerical finite volume simulations, these terms appear explicitly in the Lagrangian framework (artificial viscosity) or implicitly in the truncation terms of nonoscillatory Eulerian methods.

In this sense, I disagree with Nishikawa. Not all the truncation terms are error.

I do agree with Denaro that the integral form is more physical and the PDEs are the approximation. No physical measurement has ever been performed at a mathematical point.

The counterpart of the appearence of dissipation in the integral (or finite-lenght) formulation can be deduced from the smoothing behavior of the corresponding exact transfer function. This way, the numerical error is a superimposition on the natural smoothing of high resolved frequencies.

The most fundamental form is the integral one as it corresponds to the conservation the actual equations express. This is reflected by the fact that this formulation allows for discontinuities/irregular solution we all know as shocks. In order to go to the differential form, one has to assume regularity of the solution.

Any finite volume (fvm) formulation is in fact first of all based upon this integral form, whereas finite differences (fdm) are based upon the differential form. For regular solutions, we can usually find equivalence between the two methods, at least in some respect, allowing to analyse fvm in the same way as fdm. As the differential form is easier to manipulate, it is often used as the basis for the discussion.

The differential and integral forms have their own advantages. Finite difference method is used for differential form and finite volume method is used for the integral form. Conservation is directly taken care in integral form and for flows with shocks, the Rankine-Hugoniot relations are satisfied.

Kishore Kumar Sankara Thanks for the answer. What are the different advantages and disadvantages? besides treatment of shocks? Are there certain problems you would favour one approach vs the other?

read this article

https://math.mit.edu/classes/18.086/2014/reports/HenrikSchmidtDidlaukies.pdf

The difference b/w non-conservation (NC) and conservation (C)in terms of the equation is that the left-hand side only in Navier stroke equation i.e in NC substantial derivative of density whereas in C it is a local derivative of density and divergence of velocity component in x-direction. if you go with the theory part, kindly check from the book: https://www.eng.auburn.edu/~tplacek/courses/fluidsreview-1.pdf

These equation are basic back bone to solve fluid flow problem

I would highlight that the "conservative form" exists also for the differential equation and is also denoted by "divergence form". The non-conservative form is associated to the quasi-linear expression of the equations.

Note that the variable resolved in the differential and in the integral formulations different, this latter being associated to a local averaging, i.e. a smoothing of components in the wavenumber space.

Both equations lead to the same result. In my opinion the integral equation is easier to solve numerically.

Harald Fessler

no, your statement is not fully correct

each for has own advantages and has specific application, I guess there is priority for using them but the differential form is more accurate than Integral form, this a long story that is clarified in most of reference books like White and Anderson.

Both approaches, give their own merrits in an specific applications, but for the accuracy point of view the differential form will be more appropriate than integral form. A few references can be followed I e White and Anderson & article suggested bu Ali Jalayeri.

There is no theoretical ground to state that the differential form is more accurate!

Magnus Svärd

What do you mean? Could you elaborate ?

Magnus Svärd

Sure, it is but that is not the sense of my previous statement.

To be more detailed, I can refer to the textbook of Leveque about FVM for hyperbolic equations. You will see introduced a differential form but the solution is actually a local averaged variable.

Magnus Svärd

yes, the local averaged variable is an approximation of the strong solution. Actually it can be O(h2) for centred values and O(h) otherwise. Furthermore, using a deconvolution you can increase the accuracy to O(hk).

The second part of your answer is not clear to me. The local volume averaging is nothing but a spatial filter. You can easily see that the transfer function is a smooth one (like sin x/x). Thus, high resolved wavenumers components of the flow are smoothed. There is no need at all to introduce an artificial diffusion.

Magnus Svärd

"Any damping comes from the approximation of the fluxes along the edges. For example, on a Cartesian grid with central fluxes, the FVM scheme is identical to the finite difference scheme for the divergence form which contains no damping."

For a long time such a statement has been followed. Only for few specific case, you have the same algebric coefficients but in general no, you have FD and FVM largely differ each other.

And to understand the resulting smoothing, just introduce a generic Fourier components f(x) = F(k) ei*k*x in the definition of the local averaging

f_av(x)=1/h Int [x-h/2;x+h/2] f(x') dx'

and express the transfer function G(k).

You will understand that the smoothing is mathematical not a numerical effects of the flux reconstruction. What you have introducing a discretization of the fluxes is a numerical approximation of the exact local averaged variable. Again, if you perform correctly a FVM you have a smoothing without any upwind-like flux reconstruction.

If you want more details, you can read

Article What does Finite Volume-based implicit filtering really reso...

Magnus Svärd

Indeed what you wrote is not what a FVM would perform. The integral equation writes as

d/dt 1/h Int [xi-h/2;xi+h/2] u dx + [F[u(xi+h/2)] - F[u(xi-h/2)]]/h = 0

If the flux reconstruction is linear (second order accurate) then you have the semi-discrete

d/dt 1/h Int [xi-h/2;xi+h/2] u dx + [ F[0.5*(u(xi)+u(xi+1))] - F[0.5*(u(xi)+u(xi-1))]]/h =0

or

d u_av/dt + [ F[0.5*(u(xi)+u(xi+1))] - F[0.5*(u(xi)+u(xi-1))]]/h =0

In conclusion, only assuming uav= u +O(h2) the FVM equation is closed. Such approximations can drive to produce the similarity you cited above only for the specific linear case. (Have also a look to the FV chapter in Ferziger, Peric and Street textbook).

But the resolved variable in the integral form is ,formally, u_av, that is a (smoothed) approximation of the strong solution u.

There is also a very recent paper of van Leer and Nishikawa you can read to find further observations about FD and FV discrepancy

Article Towards the Ultimate Understanding of MUSCL: Pitfalls in Ach...

Hope that can clarify the issue.

Magnus Svärd

I wrote the continuous integral equation, uav and u are the continuous functions not the numerical approximations. The equation I wrote is the only expression the Leveque addressed while introducing the FV method, see in the textbook Eq.4.2. This is not one among different schemes but it is the exact starting equation for FVM. Also at page 66 you can read some interesting statement. Again, be careful in differencing q and Q. You will se that the flux (4.5) depends on q and the equation is not closed.

As far the concept of smoothing in FV, I want to highlight that the local average function can be written in the Fourier space as fav(k)=sin(k*h/2)/(k*h/2)*f(k) so that you see that fav(k)->f(k) for h->0. That is nothing but the mean theorem value.

If you are aware of the LES phylosophy you will get easily the concept.

However, I provide the references you can read to understand that u_av in a real FVM must know that is a local averaged value. The confusion is always a consequence of the fact that u_av and u are mathematically (not numerically) the same at second order of magnitude h. And note that h is the volume size, not necessarily equal to the grid size.

I suggest also this paper by Len G Margolin

Article Discrete regularization

If I can add a slightly different perspective, I think the point is: given the Reynolds transport theorem as the starting point of a physically sound derivation of a transport equation (say, for density with given velocity field), there is no finite scale (the only practically relevant ones) at which you can close the convective term, given the fact that it is an integral evolution equation for a volume averaged quantity (but the non averaged one is required on the faces of the volume). In this respect, any limit h->0 changes this very feature of the equation, which exists at any finite scale. From this point of view, the correspondence between any FD and FV discretization is purely accidental, even on perfectly uniform, cartesian grids.

Magnus Svärd

if we don't agree about the basic in the formulations of the conservation laws then there is nothing we can agree about the numerical analyisis.

Integrals and derivatives are discretized but that is not an alien world, the basic in the numerical analysis is to understand that your numerical solution is, actually, an exact solution of a continuous modified integral/differential equation.

Thus, the issue if the ME is consistent to the original equation in the continuous sense.

Paolo Lampitella provides a correct framework from the Reynolds theorem. And there is no numerical algorithm.

Actually, when I wrote accidental I was seriously inaccurate. It is very clear when the two discretizations, FD and FV, may (but not necessarily will) coincide: it is when spatial derivatives commute with the volume integral. Thus, another point of view is: there is no finite scale h where, in general, the two operators above commute.

Paolo Lampitella

we should realize that is much more simple to discuss about complex theories rather than about simple but basic theoretical assumptions that people interpreted in wrong way and were frozen in the years.

Magnus Svärd

this discussion is irreal!

"To derive the modified equation one has to assume that there exists a sufficiently smooth solution and it is little more than a consistency check that also provides some indication of what properties the scheme may have."

and you know that this assumption of a regular solution is necessary to write any kind of FD formula. If you don't accept that, then you cannot introduce the FD method (as well as other discretization) and you cannot have a numerical solution to prove anything.

Finally, the viscous equations (NS) have the derivative in the diffusive flux even if you use the weak form. You should rather address the perturbed Euler equations to discuss about that. The Reynolds theorem in its general form is not a PDE but, being no spatial derivative in effect, is an ODE written for the material volume.

Again, you don't agree that FD and FV are discretization of different continuous formulations. There is nothing to add.

Magnus Svärd let me first clarify that I was just trying to reword something in the hope to have it more clear what a certain point of view is (sometime I brag about being able to do this).

Maybe, I can try again (but excuse any possible naivety) by noticing that it is, indeed, in the derivation of the model that we already diverge. In the sense that, from a FV perspective, we stop before the differential PDE. The Reynolds theorem is our PDE (honestly, I didn't mention NSE because that's a very slippery territory).

And even before any discretization, it already has a closure problem, requiring on volume faces a quantity for which you don't have an equation (without further assumptions). It's not that you know a variable in the cell center but you need it on the face. You need the point variable but you evolve the space averaged one.

This, in my opinion, is very different, at the ideological level, from applying a FV discretization to a PDE in differential form.

Yet, it still appears very clearly when accuracy higher than 2nd is required for a FV discretization.

I'm not debating, at all, on which form is correct or not, better or not, etc., I'm not even equipped for that. I don't even want to say that this is the mainstream view on FV. I'm just claiming that there is a way to look at FV that has almost nothing to do with discretization and its not because of a confusion of what is what.

Magnus Svärd

"The standard FV scheme is derived from the integral of the divergence form of a conservation law on a fixed volume in space...

...This model, in divergence form, is then discretized by e.g. the FVM method"

This, I think, is where we don't agree then. I'm not claiming that your statement isn't true (of course), but that you can indeed directly discretize an integral conservation equation that was derived without ever passing trough the divergence form.

Also, I'm not that naive to not recognize that then, in a form or another, in my actual FV discretization of the equation, I may use any form of assumption on the regularity, even directly using the divergence theorem here and there.

But keeeping the discretization and the starting model equation separate has, in my opinion, some advantages.

But then, I guess, this is probably a controversial view on FV.

There is no possible controversial, any textbook intoroduce the FV as a discrete form of the conservation law wherein the integral surface of the fluxes is discretized. The use of the divergence is just a misleading way to talk about FVM.

Leveque uses the differential form applied on the averaged variable. See Pages 65-66.

Magnus Svärd

However, if you divide space into finite volumes and take phi_j to be the functions that are 1 on the volume j and zero elsewhere you get the base form of a finite volume scheme. Of course, you need to approximate the boundary integrals to get one particular instance of an FVM scheme

that is exactly what we are assuming, as reported also in the Leveque textbook. He demonstrates that the physical conservation law in integral form is also one weak form for a specific choice of the test function. All other test functions produce a different FEM framework. FVM is nothing but a specific form of FEM.

The disagree is in the fact the FVM are the discretization of the surface integral of the fluxes as represented in the Reynolds theorem.

And the flux function is F(u) while the time updating is for u_av. The integral equation, if the unknown is the local averaged variable, is not closed. No matter about divergence form or not. What you do in a FEM manner is the finding of the solution of the function u under the integral. You are not considering the difference in the u_ac since you are assuming convergence at h->0. But there is no h going to zero.

Magnus Svärd you have been crystal clear, and I agree to everything.

Also, I think I perfectly understand now your point of view, which (bragging again on my capability of conciling things) I would roughly resume as:

"Nice and all, but you are getting a solution to what? If you don't converge with h->0, is that even a solution? And if you do, why in the hell are we still talking about this?"

To these questions, honestly, I'm not sure if I have an answer.

Nonetheless, I think I could also reformulate my point of view (which might become less mainstream, I understand) using exactly the distinction between model and discretization that I was talking about before: what if I want exactly the solution for volume averaged variables for certain, FIXED, volume sizes? That is, I want to know the evolution of volume averaged quantities on certain volumes of fixed size that I decided in advance.

I hope it is clear, in this case, that my finite volumes are not anymore allowed to shrink, otherwise the equation is formally changing. Also, as a matter of fact, when we convert a transport equation in integral form to its discretized counterpart, the only approximation needed is in the fluxes for the faces, not the fact that there are finite faces and volumes.

Magnus Svärd I feel that it is my colloquial English that failed the most here as, to me, the doubts you advanced seem, indeed, to match exactly those that I colloquially phrased, in what had to be considered a rethorical tone, given the context.

Because, nowhere I wrote that I am willing to use a non convergent code, nor I said I want to use any given FV code to get a one shot solution for a fixed grid.

What I did was a further attempt to clarify what I wanted to highlight about FV with a limit, absurd, example question that, honestly, I really tought could have worked (but appraently not).

Never I said that a FV discretization is better than FD (despite having a clear personal preference for the case of unstructured grids, due to the resulting programming simplicity of classical FV), nor that FD cannot be conservative, nor that, given the proper conditions, the two cannot converge to the same mathematical solution of what, in the end (pun intended), is the same mathematical problem.

Nor, allow me, I ever considered that you don't know what you're talking about. In contrast, trying to fail me like a numerical analysis student, who might or not know certain stuff, I feel, is preventing you to even try to understand what I want to say.

Which, in the end, is not even controversial at all. Just recognizing that at any finite scale (h>0) the discretized integral equation is ALSO related to the evolution of a volume averaged quantity which is not the same as the point quantity, with the two being apart (at best) by O(h^2). I mean, this is not someting I came up with, it has been shown repeatedly by the most diverse people around the world (including those trying to achieve higher than 2nd order accuracy with FV).

I would accept if you said that this is so obvious to be irrelevant from your point of view, but not that this is false, honestly. Of course, it is important to me but, you can backcheck, I never wrote that I wanted this to be recognized as universally relevant.

Magnus Svärd yes and no. You didn't misunderstood that in the literal sense. But you did in recognizing those as just hypothetical question, which I put forward only as a mean to put the thing under a still different angle. The idea was that my colloquial resume, which I would rephrase like this now:

Magnus: "What you are writing is nice and all but, what continuous equation are you solving with your FV method? If you don't converge with h->0, is the numerical solution you obtain even a solution to some equation? And if you do actually converge, then why in the hell are we still talking about this, aren't we doing the same thing then?"

Paolo: "Yes indeed, I just use a FV method that converges with h->0 and, for all practical means, I always consider my solution as the one for the weak problem, and the one I consider as my final solution is indeed the last of a series performed on increasingly refined grids. So, in the end, I recognize that everything you have been saying is correct and this might seem a waste of time, because we are indeed doing the same thing. Yet, I am unable to not recognize that any solution for finite h has, indeed, an additional meaning for a volume averaged quantity whose error is solely dependent from the flux discretization. What is the meaning of this NUMERICAL solution for finite h, considering it is not obtained as a limit process? I don't have a clear answer for this, at least not in the frame we are discussing here"

was sufficient to clarify that I didn't want the following question to be meant literally. Obviously I was wrong, and it is certainly my fault.

However, let me just add that, EVEN IN THE PURELY HYPOTHETICAL CASE that I was serious about keeping the volumes fixed in a practical computation (which I hope is now clear is NOT what I ever meant), nowhere I wrote that the numerical error shouldn't be reduced in some other way to achieve convergence in this purely hypothetical case. As this was purely hypothetical, I had nothing specifically, or even achievable, in mind. Yet, p-refinement is something I would probably look at if I have to.

Besides this, I also agree on the fact that there is nothing left to add here. Thank you for the discussion as well.

Such an irreal discussion now...

1) Assume a finite lenght H of the volume is different from the computational grid size h and prove to which kind of variable you converge for h->0 while H is constant. You can build a subset of FVs going to zero volume measure and still have an equation for resolving the volume averaged variable.

2) What about initial and BCs for the differential and integral problems? To be well posed, to be congruent to the meaning of the variables you wanto to solve, what about the mathematics of the problem?

3) What if I have an analytical solution and I want to test the convergence order (the slope of the error) for a IV order accurate FD and a IV order accurate FV?

After that, I thank you for the responses and I think we can quit in discussing each other.

@Paolo Lampitella Despite the intransigence of some responders, I think your posts are very clear. I am in substantial agreement with you (and Filippo) and have published several papers describing modifications to the equations when you require that the control volume remain finite. I have gone even further, and asserted that it is the discrete equations that best describe our measurements of nature, and that the PDEs are the approximation.

I would note that most interesting problems in fluid flows cannot be solved analytically, either in integral form or as PDEs. It is then necessary, and usually very effective, to solve those problems on the computer. You wrote that as you change the size of the finite volumes you change the equations. I would add that necessarily you also change the solutions. This simply highlights the important ideas of Heisenberg and Bohr, namely that descriptions of nature cannot be independent of the measuring process. The size of the computational cell is an analog of the measuring device.

The notion of convergence is a chimera at least in the case of high Reynolds number flow. There we solve regularized Euler equations whose only length scale is the cell size. Then the statement h --> 0 is nonsensical. h is a length scale and can only be deemed large or small when compared with other length scales.

You can find many of my papers on research gate that give substance to these ideas; I would particularly recommend "Finite scale theory: the role of the observer in classical fluid flows." If you have papers of your own to recommend, please respond or message me.

Dear Len G Margolin , thank you very much for the sanity check. I would lie if I said that I don't know most of your works related to the subject. Unfortunately, life after Ph.D. has made it extremely difficult for me to dig deeper in most papers (if any at all), so I only kept track of your work, but will certainly take the chance now.

Nonetheless, while my academic production has been irrelevant, your works on the subject of which I was aware at the time (e.g., Article Finite-scale equations for compressible fluid flow

Thesis Large Eddy Simulation for Complex Industrial Flows

I don't know if I would reccomend reading it (as today I recognize it to be long and pedantic even for myself), but I would certainly be glad if you did.

In passing, I can't resist to add that, indeed, at any finite scale, our "controversial" reasoning naturally leads to the appearance of additional terms in the continuity equation.

Hi dear.. Please follow the book; Fluid Mechamics by FM White. Both forms are explained in detail with relevant geometry.

Len G Margolin

Dear Len,

I appreciate that you highlight in this discussion the concept of computing numerically always at a finite size. One of the conceptual conflict in the convergence analysis is the real meaning of what we assume for h->0.

Magnus Svärd

Dear Magnus, I don't think there is any doubt about the literature (please consider also Richtmyer, von Neumann, Morton as well as Marchuk and other russian people) to consult. The issue is here about what you are stating and what other people are stating. And that is not a competition but a scientific discussion. The readers will have their opinion about that.

Paolo Lampitella

Dear Paolo,

discussing is the best way to think and to grow. Don't be surprised by new intepretations of topics that are considered "classic". Often a progress in science is based on the fact that nothing is untouchable. Feynman highlighted the role of the doubt in the scientific progress.

Filippo Maria Denaro

Dear Filippo

as this conversation is drawing to a close, perhaps I may offer a slight diversion.

You mention the name Richtmyer, and most people are familiar with the classic book by Richtmyer & Morton on Difference Methods. Many people will also recognize him as the co-author of the 1950 paper with von Neumann which introduced many of the fundamental ideas of CFD, including artificial viscosity and Fourier stability analysis. Richtmyer worked at Los Alamos and von Neumann was a frequent consultant there.

There are unpublished reports written mostly around 1947-48 that are the foundation of the 1950 paper. They show that the idea of artificial viscosity, and in particular the unexpected form that is quadratic in the velocity gradient, were the sole work of Richtmyer. In fact, that form is derived by Richtmyer in his reports, although it is only presented fait accompli in the paper.

Richtmyer is also famous in some communities for his theoretical derivation of the Richtmyer--Meshkov instability.

Von Neumann's star was so bright that it often eclipsed the work of his collaborators. But Bob Richtmyer was a true hero of CFD.

Differential form is useful in CFD, integral form is more useful in control volumes.

The Finite Difference Method (FDM) uses the differential form of the conservation equations as its starting point, whereas the Finite Volume Method (FVM) employs the integral form of the same equations as its starting point. As you may know, Navier-Stokes (NS) equations are mathematical expressions of the conservation laws (i.e. conservation of mass, momentum and energy). The strong conservative (i.e. all terms have the form of divergence of a vector and a tensor) form of NS equations, when used together with FVM, automatically ensures global momentum conservation in the calculation. While the non-conservative form of NS equations is often used in FDM. In the limit of a very fine grid, both equation forms and numerical solution methods give the same solution. For further reading, the following textbook on CFD may be very helpful:

Computational Methods for Fluid Dynamics by Joel H. Ferziger & Milovan Peric, 3ed,, Springer, 2002.

Academic Resources on Kinematics and Conervation Laws for Classical Fluid are provided in :

Presentation ECFM-Chapitre 1 : Cinématique du Milieu Continu

Presentation ECFM-Chapitre 2 : Lois de Conservation

Presentation ECFM-Chapitre 3 : Equations Générales du Fluide Classique

Jamel Chahed what about the material you linked and the discussion?

Filippo Maria Denaro The three chapters proceed from an integral formulation to lead to the formulation of local differential equations and the constitutive laws of the classical fluid. If you find this unrelated to the discussion, well, pretend they don't exist

Jamel Chahed

the theoretical framework for the two formulations is a classic topic illustrated in fundamental textbooks.

Here the discussion is differently focused. Thus, which of your three files is pertinent to the discussion in such a way that a reader can be specifically addressed?

Filippo Maria Denaro Quote "the theoretical framework for the two formulations is a classic topic illustrated in fundamental textbooks". So What ? This is an other fundamental textbook, and I pretend it has an original construction. Quote "Here the discussion is differently focused. Thus, which of your three files is pertinent to the discussion in such a way that a reader can be specifically addressed?" If, as you mentioned "the discussion is differently focused" thus none of the three files is pertinent to the discussion. Would you like I remove tem ?Aditya GhantasalaAditya Ghantasala

Jamel Chahed

I would like you give a specific contribution to the discussion, if you have. Again, which one of you your file is relevant to this discussion ? No need to remove, just address the part to read.

Thank you Dear Colleague Filippo Maria Denaro for your concern and for your generosity in allowing me to maintain my answer to the present question. I thought indeed, that these three chapters are formulated in the spirit of the question posed; which concerns continuous media formalism fundamentally based on the passage from the integral formulation to the local one, in order to establish the differential formulation balances. Correlatively, the constitutive relations are formulated from the principles of thermodynamics and applied to the classical fluid with constant properties. It is for this reason that the three chapters are complementary.

This formalism is also the basis for the formulation and study of single and multiphase laminar and turbulent flows in nature and engineering applications: physics of turbulence and modeling, two-phase flows and two-fluid models, dynamics and thermodynamics of geofluids, atmospheric dispersion. Academic resources and tools related to these areas of Applied Fluid Mechanics are brought together in the project and made available to student and researchers:

https://www.researchgate.net/project/Single-Phase-and-Multiphase-Turbulent-Flows-SMTF-in-Nature-and-Engineering-Applications

Jamel Chahed

the original post has a clear keyword that is CFD. Previous answers already fully addressed the theoretical role of the mathematical formulation in terms of divergence, quasi-linear and integral form of the governing equations. No matter about that, it is their usage in numerical simulation the goal of the discussion.

Anyway, I suppose your contribution would be better promoted and reach more people if written in english.