[TurbJet]

Hi,

I've been reviewing the basic idea of FVM. However, there are few questions that I am confused. Lets take simple 1D linear advection for example

1. After the interpolation and obtain the values of q at the cell faces, e.g., , we need to assemble flux. In Ferziger's book (Computational methods for fluid dynamics), he directly assemble the flux using the face values, i.e.,. However, in LeVeque's book (Finite volume method for hyperbolic problems), after the interpolation (or what he calls "reconstruction"), he still uses some numerical flux function F to approximate the time-average flux at the faces (e.g., Lax-Wendroff, Lax-Friedrichs). I am confused by LeVeque's approach: since we already know all the values at cell faces (after reconstruction), why still need the numerical flux? Why can't we directly use those values to compute the flux (like Ferziger's way)? According to LeVeque, the numerical flux is an approximation of the term (in which the flux for the linear advection above). But I am not sure if this is the answer to my question
2. What exact does reconstruction (as in LeVeque's book) mean? Is it just the interpolation for obtaining the face values?
3. According to Leveque, the solution variable is actually an approximation of the integral, i.e., . Is this approximation just the "piecewise constant reconstruction"?

[FMDenaro]

There is no conflict between the two textbooks. Leveque explains clearly the FV formulation in Eq.(4.1)-(4.6). You have to integrate over a finite volume and integrate further in time.

In the textbook of Ferziger you have to consider the section of the FVM starting from the integral form and you will see the same formulation.

[TurbJet]

Originally Posted by FMDenaro

There is no conflict between the two textbooks. Leveque explains clearly the FV formulation in Eq.(4.1)-(4.6). You have to integrate over a finite volume and integrate further in time.

In the textbook of Ferziger you have to consider the section of the FVM starting from the integral form and you will see the same formulation.

Yeah, I saw that. I am just confused, as I said, we already have face values (from interpolation), why don't we directly assemble the flux use the face value? Why we need the numerical flux?

[sbaffini]

Roughly speaking, because LeVeque has to teach the whole thing also for non linear cases and for a very large family of equations. Ferziger mostly treats linear equations (with the mass flux already given), while the determination of the mass flux is made only with a specific implementation and equation in mind. Besides this, Ferziger uses the method of lines, while LeVeque mostly doesn't.

[TurbJet]

Originally Posted by sbaffini

Roughly speaking, because LeVeque has to teach the whole thing also for non linear cases and for a very large family of equations. Ferziger mostly treats linear equations (with the mass flux already given), while the determination of the mass flux is made only with a specific implementation and equation in mind. Besides this, Ferziger uses the method of lines, while LeVeque mostly doesn't.

That makes sense. So can I presume that they are two different approaches to FVM?

Plus, is there any other higher order numerical flux? LeVeque extensively talks about Lax-Wendroff and Lax-Friedrichs, which only 2nd-order.

[vesp]

Originally Posted by TurbJet

That makes sense. So can I presume that they are two different approaches to FVM?

Plus, is there any other higher order numerical flux? LeVeque extensively talks about Lax-Wendroff and Lax-Friedrichs, which only 2nd-order.

yes, read up on ENO/WENO schemes. Chi Wang Shu comes to mind...

[FMDenaro]

As a general rule the flux reconstrunction means:


- assuming that the averaged values Q are computed in each FV at time t0;
- considering that the derivative in time dQ/dt depends on the difference of the fluxes on the faces, the fluxes depending however on q not on Q.

- the reconstruction of the fluxes requires the time integration between t0 and t0+dt of the difference function F[q(Q)]i+1/2 -F[q(Q)]i-1/2.
- high order accuracy means you have to implement both a high order time integration and a high order reconstruction q(Q) (like in ENO/WENO/TENO).


Leveque provides many details for the one-dimensional equations (linear and non-linear) as well as for one-dimensional system of equations.

Note that an example of high order reconstruction in 2d space is illustrated in the textbook of Ferziger but you can also see as further example the original QUICK/QUICKEST schemes of Leonard.

[FMDenaro]

Originally Posted by TurbJet

Yeah, I saw that. I am just confused, as I said, we already have face values (from interpolation), why don't we directly assemble the flux use the face value? Why we need the numerical flux?

No, we don't have the face values for the flux as it would require the knowledge of the pointwise function q but we know the averaged function Q. Therefore, "reconstruction" is not a simple interpolation but the introduction of an approximation for the functional relation q=q(Q).

[praveen]

If you reconstruct a unique value at each face (usually with a central stencil) then you use that to compute the flux. This would lead to central difference type schemes.

For convection dominated problems, say at high Re, you may want upwind schemes. You reconstruct two values at each face, one with a left biased stencil and another with right biased stencil. Then to compute the flux, you will need a numerical flux, which is the subject of LeVeque's book.

[TurbJet]

Originally Posted by praveen

If you reconstruct a unique value at each face (usually with a central stencil) then you use that to compute the flux. This would lead to central difference type schemes.

For convection dominated problems, say at high Re, you may want upwind schemes. You reconstruct two values at each face, one with a left biased stencil and another with right biased stencil. Then to compute the flux, you will need a numerical flux, which is the subject of LeVeque's book.

Originally Posted by FMDenaro

As a general rule the flux reconstrunction means:


- assuming that the averaged values Q are computed in each FV at time t0;
- considering that the derivative in time dQ/dt depends on the difference of the fluxes on the faces, the fluxes depending however on q not on Q.

- the reconstruction of the fluxes requires the time integration between t0 and t0+dt of the difference function F[q(Q)]i+1/2 -F[q(Q)]i-1/2.
- high order accuracy means you have to implement both a high order time integration and a high order reconstruction q(Q) (like in ENO/WENO/TENO).


Leveque provides many details for the one-dimensional equations (linear and non-linear) as well as for one-dimensional system of equations.

Note that an example of high order reconstruction in 2d space is illustrated in the textbook of Ferziger but you can also see as further example the original QUICK/QUICKEST schemes of Leonard.

I went through Qiqi Wang's class at MIT (on YouTube). His explanation of reconstruction is (if I understand him correctly) "directly interpolate the flux", rather than interpolate the face value then compute the flux (as what I thought). How do you think about it?

[Eifoehn4]

Dear all,

i am a little bit confused nobody mentioned the Riemann problem theory (exact or approximativ), which is the most intuitive way to explain finite volume methods. Not only from a mathematical point of view but also from a physical one.

If you succeed to understand what a Riemann problem is you won't have any problems to understand how finite volume methods work.

Regards

[FMDenaro]

Originally Posted by Eifoehn4

Dear all,

i am a little bit confused nobody mentioned the Riemann problem theory (exact or approximativ), which is the most intuitive way to explain finite volume methods. Not only from a mathematical point of view but also from a physical one.

If you succeed to understand what a Riemann problem is you won't have any problems to understand how finite volume methods work.

Regards

The Riemann problem at each face is one of the methods for the flux evaluation, starting from the hystorical Godunov method then developed by Roe.However, developing a high-order and multidimensional FVM based on Riemann solver is cumbersome.

Furthermore, the typical field of application of Riemann solvers is for Euler equations where singularity (shock) appears. A different field is the Navier-Stokes equations at very low Mach (or incompressible assumption) where the flux reconstruction is based on different ideas and the pressure has not thermodynamics meaning.

[Eifoehn4]

Originally Posted by FMDenaro

The Riemann problem at each face is one of the methods for the flux evaluation, starting from the hystorical Godunov method then developed by Roe.However, developing a high-order and multidimensional FVM based on Riemann solver is cumbersome.

Furthermore, the typical field of application of Riemann solvers is for Euler equations where singularity (shock) appears. A different field is the Navier-Stokes equations at very low Mach (or incompressible assumption) where the flux reconstruction is based on different ideas and the pressure has not thermodynamics meaning.

Your remarks are entirely correct.

Nevertheless, when the question arises about the motivation of the flux approximation in the context of finite volume methods then it is hard to give a vivid answer without this theory, especially if someone has not understand that the physical flux at the face boundaries is generally not unique with finite volume methods.
Note that the Riemann problem theory does not only exist for compressible flows. For example the exact Riemann problem for a scalar transport equation is the upwind flux approximation. This will give you directly an answer why a central approximation is not recommended here.

Beside the common compressible and incompressible Navier Stokes equations there exist other identical physical models which are purly hyperbolic (at least for the compressible one) and may completely be explained by the generalized Riemann theory, which does not neccessaraly be a constant initial value problem.

Regards

[sbaffini]

Originally Posted by Eifoehn4

Your remarks are entirely correct.

Nevertheless, when the question arises about the motivation of the flux approximation in the context of finite volume methods then it is hard to give a vivid answer without this theory, especially if someone has not understand that the physical flux at the face boundaries is generally not unique with finite volume methods.
Note that the Riemann problem theory does not only exist for compressible flows. For example the exact Riemann problem for a scalar transport equation is the upwind flux approximation. This will give you directly an answer why a central approximation is not recommended here.

Beside the common compressible and incompressible Navier Stokes equations there exist other identical physical models which are purly hyperbolic and may completely be explained by the generalized Riemann theory, which does neccessaraly be a constant initial value problem.

Regards

You are absolutely right as well. But I'm not sure that your line would have helped to put Ferziger approach in context. What you are basically saying is "never mind Ferziger, just follow LeVeque". Legit, but the original question mentions Ferziger explicitly.

[Eifoehn4]

Yes for a beginner or a more advanced CFD enthusiast i would recommend to not start with Ferziger book.

[FMDenaro]

Originally Posted by Eifoehn4

Your remarks are entirely correct.

Nevertheless, when the question arises about the motivation of the flux approximation in the context of finite volume methods then it is hard to give a vivid answer without this theory, especially if someone has not understand that the physical flux at the face boundaries is generally not unique with finite volume methods.
Note that the Riemann problem theory does not only exist for compressible flows. For example the exact Riemann problem for a scalar transport equation is the upwind flux approximation. This will give you directly an answer why a central approximation is not recommended here.

Beside the common compressible and incompressible Navier Stokes equations there exist other identical physical models which are purly hyperbolic (at least for the compressible one) and may completely be explained by the generalized Riemann theory, which does neccessaraly be a constant initial value problem.

Regards

You are right if we consider only some specific aspects of the FVM that apply only to some cases, in particular to hyperbolic problems. For example the textbook of Leveque is well suited for such field.



1) central flux reconstruction is well suited for the incompressible form of the NSE, it is quite a standard in a lot of hystorical FV codes. The momentum equation is parabolic and the continuity is hyperbolic but is solved by trasforming in the elliptic equation.

2) The use of Riemann solver in a FVM is based on a discontinuity of values between the left and right faces but that says nothing about the way the averaged function behaves within each FV. It could be piece-wise constant or linear or quadratic and so on. This part is a key in the flux reconstruction out of the Riemann problem. As you wrote, we cannot express a unique primitive function from a known averaged value.
3) developing a high order accurate FVM base on the Riemann solver (more than the simple first order upwind in the Godunov method) is quite problematic in multidimensional flows.


I think that is better to analyse differently FVM either for hyperbolic Euler equations or parabolic/elliptic NSE equations. Some standard CFD textbooks illustrates the FVM without considering at all the Riemann solver in the flux reconstruction.

[FMDenaro]

Originally Posted by Eifoehn4

Yes for a beginner or a more advanced CFD enthusiast i would recommend to not start with Ferziger book.

What book do you think is better for that?


From my experience as teacher in the CFD field is quite problematic to introduce the FVM starting with the Riemann solver-based flux reconstrution. You need first a background in gasdynamics.

[vesp]

Why would multi D Riemann solvers be important? all the FV codes I have seen in use at big conpanies just use 1D RP solvers normal to the faces...

[TurbJet]

Originally Posted by vesp

yes, read up on ENO/WENO schemes. Chi Wang Shu comes to mind...

Yeah, I know the ENO family. But from what I read (Chi-Wang Shu, ENO and WENO for hyperbolic conservation law, ICASE, 1997), under FVM framework, ENO/WENO only reconstruct the left/right face values, which still need some numerical flux to approximate the physical one. And Shu recommended 3 numerical fluxes in this paper: 1. Gudonuv flux; 2. Enquist-Osher flux; 3. Lax-Friedriches flux.

From this paper, seems like only under FDM framework will ENO/WENO directly go ahead and reconstruct the fluxes.

[Eifoehn4]

Originally Posted by FMDenaro

What book do you think is better for that?


From my experience as teacher in the CFD field is quite problematic to introduce the FVM starting with the Riemann solver-based flux reconstrution. You need first a background in gasdynamics.

Yes that's maybe true, but i think it is also hard to skip this topic when someone starts to learn the finite volume theory and you have to explain him afterwards e.g. things asked by TurbJet.

In my opinion it is really helpful to introduced common finite volume
approximations along with the Riemann problem theory.

The books of Leveque and Toro are good references, but i would also suggest to use second literature, where the focus is more on numerical methods, especially for the incompressible Navier Stokes equations.

Regards