[granzer]

What are the differences between Conservative differential form, Non conservative differential form, Conservative Integral form and Non conservative integral form of differential equations?I know that there are a lot of different post on 'conservative vs non conservative form' and ' differential vs integral form' but from what I have read conservative and non conservative form of the governing equations also have differential and integral type. Where are these different types of equations applied and what are the advantages and disadvantages of one over the other ?

[FMDenaro]

Originally Posted by granzer

What are the differences between Conservative differential form, Non conservative differential form, Conservative Integral form and Non conservative integral form of differential equations?I know that there are a lot of different post on 'conservative vs non conservative form' and ' differential vs integral form' but from what I have read conservative and non conservative form of the governing equations also have differential and integral type. Where are these different types of equations applied and what are the advantages and disadvantages of one over the other ?

There are a lot of discussions about this topic. I suggest also to have a reading to the Chap.1 of the Hirsch textbook that addresses the topic in a clear way.
Briefly, I use the mass equation to address the differences:


- Integral conservative form


d/dt Int [V] rho dV + Int [S] n.v rho dS =0

- differential conservative form


d rho/dt + Div(v rho) =0


- differential quasi-linear (non conservative) form


d rho/dt + v.Grad rho + rho Div(v)=0



Note that conservation property is valid for the physical variables that can be conserved, that is mass, momentum and total energy.


For example, you can write the integral form of the kinetic energy equation but this variable is never conserved in the general case.

[granzer]

Originally Posted by FMDenaro

There are a lot of discussions about this topic. I suggest also to have a reading to the Chap.1 of the Hirsch textbook that addresses the topic in a clear way.
Briefly, I use the mass equation to address the differences:


- Integral conservative form


d/dt Int [V] rho dV + Int [S] n.v rho dS =0

- differential conservative form


d rho/dt + Div(v rho) =0


- differential quasi-linear (non conservative) form


d rho/dt + v.Grad rho + rho Div(v)=0



Note that conservation property is valid for the physical variables that can be conserved, that is mass, momentum and total energy.


For example, you can write the integral form of the kinetic energy equation but this variable is never conserved in the general case.

Thank you. Yes, I know if we are talking about the conservative form then the equation has the divergence of flux (here for mass conservation it's the mass flux) but what does integral conservative form mean..every explanation of conservative form is using the differential conservative form and never the integral form.

[FMDenaro]

"every explanation of conservative form is using the differential conservative form and never the integral form. "


who told you that?? I addressed the textbook of Hirsch but you can also see the chapters in the Peric and Ferziger textbook, as well as other CFD books.


And in a more general fluid mechanics framework, the conservation equations are always derived from the Reynolds transport theorem that is by definition in integral form.

[granzer]

Originally Posted by FMDenaro

"every explanation of conservative form is using the differential conservative form and never the integral form. "


who told you that?? I addressed the textbook of Hirsch but you can also see the chapters in the Peric and Ferziger textbook, as well as other CFD books.


And in a more general fluid mechanics framework, the conservation equations are always derived from the Reynolds transport theorem that is by definition in integral form.

Will I be writing in saying that the integral form here is the weak form as it is "averaging" the over the control volume and so the condition of continuity is relaxed? and are ever integral equations in weak form?

[FMDenaro]

Originally Posted by granzer

Will I be writing in saying that the integral form here is the weak form as it is "averaging" the over the control volume and so the condition of continuity is relaxed? and are ever integral equations in weak form?

It can be demonstrate that the volume integral formulation for the conservation equation is a special case (in the sense of the use of specific test function) of weak formulation. You can write other weak formulations that do not correspond to the conservation laws.


Note that in case of the presence of second order derivatives (as in the diffusive terms) the integral formulation still requires the continuity for the first derivative.

[arungovindneelan]

Originally Posted by FMDenaro

For example, you can write the integral form of the kinetic energy equation but this variable is never conserved in the general case.

Why do we call a method is conservative just because that we can write it in divergence form and it is physically a conservative quantity? Why do we worry about divergence (a spatial operator)? Any conservative quantity is a function of space and time but not worrying about time!

Divergence can make the integral more accurate and easy on irregular domains. Could you give me a better reason or explanation other than that for an integral form to be a conservative scheme?

Why can't differential form can't be conservative? It is also satisfying the governing equation! In some special cases, even higher order differential and higher order integral form are reduced to the same formulation!

[FMDenaro]

Originally Posted by arungovindneelan

Why do we call a method is conservative just because that we can write it in divergence form and it is physically a conservative quantity? Why do we worry about divergence (a spatial operator)? Any conservative quantity is a function of space and time but not worrying about time!

Divergence can make the integral more accurate and easy on irregular domains. Could you give me a better reason or explanation other than that for an integral form to be a conservative scheme?

Why can't differential form can't be conservative? It is also satisfying the governing equation! In some special cases, even higher order differential and higher order integral form are reduced to the same formulation!

The divergence differential form is conservative if you compute the unique flux function. The quasi-linear differential form is not conservative.
See my answer in the other post.