[arungovindneelan]

What is conservative discretization and what are the requirements it should satisfy to be a conservative discretization? Why FVM is conservative scheme?

The integration of governing differential equation (GDE) over the cell and discretizing the equation that we call as conservative discretization



Is there is any rule that the limit should be from -0.5 to 0.5? why not -0.3 to 0.7? or 0 to 1? will it cause an error because of not having symmetricity?

How can one say this equation is conservative but FDM (differential form) is not conservative? but both of them satisfy the governing equation. Why any scheme that satisfies the governing equation cannot be a conservative method? Assuming that the equation in conservative form (mathematically conservative form).
Thank you.

[FMDenaro]

Originally Posted by arungovindneelan

What is conservative discretization and what are the requirements it should satisfy to be a conservative discretization? Why FVM is conservative scheme?

The integration of governing differential equation (GDE) over the cell and discretizing the equation that we call as conservative discretization



Is there is any rule that the limit should be from -0.5 to 0.5? why not -0.3 to 0.7? or 0 to 1? will it cause an error because of not having symmetricity?

How can one say this equation is conservative but FDM (differential form) is not conservative? but both of them satisfy the governing equation. Why any scheme that satisfies the governing equation cannot be a conservative method? Assuming that the equation in conservative form (mathematically conservative form).
Thank you.

Your question is basically a homework of a CFD student, you can find the answer in many CFD textbooks.
Start to write the integral form of the equations, you will see that FVM are deduced from this continous formulation that is conservative by definition

[arungovindneelan]

Originally Posted by FMDenaro

Your question is basically a homework of a CFD student, you can find the answer in many CFD textbooks.
Start to write the integral form of the equations, you will see that FVM are deduced from this continous formulation that is conservative by definition

I'm unable to understand, any speciality about FVM, except guess-divergence theorem so that we may able to calculate divergence relatively more accurate in the irregular mesh. From your answer, can I interpret any scheme that satisfies governing equation is conservative? If not, what are the properties that should be satisfied by a scheme to be a conservative scheme? I'm unable to understand how come one integral form made a discretization conservative? but both forms are valid differential equations.

Why do all the FVM formulations use -0.5 to 0.5? Is there is any physical significance?

Thank you.

[FMDenaro]

Originally Posted by arungovindneelan

I'm unable to understand, any speciality about FVM, except guess-divergence theorem so that we may able to calculate divergence relatively more accurate in the irregular mesh. From your answer, can I interpret any scheme that satisfies governing equation is conservative? If not, what are the properties that should be satisfied by a scheme to be a conservative scheme? I'm unable to understand how come one integral form made a discretization conservative? but both forms are valid differential equations.

Why do all the FVM formulations use -0.5 to 0.5? Is there is any physical significance?

Thank you.

Conservation means that the time variation of the volume averaged extensive variabile (mass, momentum, total energy) depends only upon the prescribed fluxes on the boundary. It does not require to invoke the divergence operator.
This principle is intrinsic in the Reynolds transport theorem and has nothing to do with a nunerical topic. It is a physical fundamental principle that we simply want to fulfill numerically.

The question about \-0.5 in the integral is not well posed. You must write the volume integral over a certain finite volume. In the integral must appear the widths of the volume.

[arungovindneelan]

Originally Posted by FMDenaro

Conservation means that the time variation of the volume averaged extensive variabile (mass, momentum, total energy) depends only upon the prescribed fluxes on the boundary.

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I like this answer because the standard "telescoping property" definition does not involve time variation, I think you have included that in this definition. Can I consider this answer as "telescoping property" that considered time variation effects?

All the conservative discretizations I read in books and paper are in the intergral form. I like to know your opinion on: can only the integral form can make the scheme conservative but not the differential form provided that the GDE is in the conservative form?
Thank you.

[FMDenaro]

Originally Posted by arungovindneelan

.

I like this answer because the standard "telescoping property" definition does not involve time variation, I think you have included that in this definition. Can I consider this answer as "telescoping property" that considered time variation effects?

All the conservative discretizations I read in books and paper are in the intergral form. I like to know your opinion on: can only the integral form can make the scheme conservative but not the differential form provided that the GDE is in the conservative form?
Thank you.

Definitely yes, the telescopic property is included in my definition. Fro steady flow it implies that the summ of all the fluxes must vanishes but in the unsteady case (as originally formulated by the Reynolds theorem), the summ of all the fluxes results in the time variation of the volume averaged variable (note how is a function only of the time for a fixed volume).


The discrete conservation property can be "extended" somehow also to the differential divergence form where the time derivative of the pointwise variable depends only on the divergence of the fluxes. Once computed a flux it must be unique for two adjacent cell. What is theoreticall questionable is the denomination of conservation for a pointiwise variable. It has no sense to define a conservation property of a quantity when you measure it on a volume of vanishing measure, that is on a mathematical point.
For this reason I always consider the integral form as the natural one for the FV discretization.

[arungovindneelan]

Originally Posted by FMDenaro

What is theoreticall questionable is the denomination of conservation for a pointiwise variable. It has no sense to define a conservation property of a quantity when you measure it on a volume of vanishing measure, that is on a mathematical point.
For this reason I always consider the integral form as the natural one for the FV discretization.

Even though the governing equation is in integral form, the solution obtained from FVM tends to GDE only the grid size approaches to zero. That also the basic requirement of a differential form. I will open another question regarding this and the accuracy of integral and differential form in some very basic ODE that has a solution. Error induced my "method of line" assumption on PDE and the effect of that on conservativeness.

[FMDenaro]

Originally Posted by arungovindneelan

Even though the governing equation is in integral form, the solution obtained from FVM tends to GDE only the grid size approaches to zero. That also the basic requirement of a differential form. I will open another question regarding this and the accuracy of integral and differential form in some very basic ODE that has a solution. Error induced my "method of line" assumption on PDE and the effect of that on conservativeness.

But you do not need to invoke the convergence of the FV-based solution towards the differential one... In other words, you can compute the weak solution without the need to invoke the continuity of the function.

Be aware that you can find in literature the differential equation but that implied the commutation of the averaging and differential operators so that the PDE acts on the locally averaged variable not on the pointiwise variable.
This is the basis on the hystorical LES approach that use the volume averaging as filter.