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.

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 see a lot of confusing answers. First off, the pde can be written in conservative (divergence) or non-conservative form. The conservative form can be discretized by either an FV or an FD scheme. (An FD scheme is _NOT_ by definition non-conservative as some answers imply.) If shocks or other discontinuous features in the solution can be expected then a conservative FV or FD scheme must be used to catch a weak solution. If the solution is smooth it is essentially irrelevant if the scheme is conservative or non-conservative. A proof of stability will ensure convergence in either cases and a high-order discretization is often preferable. Finally there are answers who seem to say that there is a profound difference in interpretation of FV and FD. That is not the case since a conservative second-order FD scheme is in fact the same as a common FV scheme. On structured grids it is easy to recast any conservative FD scheme to an FV framework. (On unstructured grids one should also be aware that many of the standard low-order FV schemes are inconsistent and therefore useless.) Furthermore, interpreting the solutions as either means or point values have little importance. A scheme that converges will converge in both the mean and pointwise a.e. as long as the solution is smooth. (The difference between a mean and a point value in the same cell is O(h).)

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.