# Roe flux scheme, why Roe's averages important

 Register Blogs Members List Search Today's Posts Mark Forums Read

 October 10, 2012, 09:16 Roe flux scheme, why Roe's averages important #1 Member   Shenren Xu Join Date: Jan 2011 Location: London, U.K. Posts: 63 Rep Power: 8 Dear colleagues, Could anyone please explain to me or point me to some reference on why Roe's averages are so important in Roe flux scheme. A simple question I have is what happens if I simply take the arithmetic average or geometric average of the left and right state, instead of strictly follow the Roe's average formulation. In fact, I remember I read in Fluent's manual that the Roe's scheme implemented therein uses simple arithmetic average, don't know why. Couldn't find the link now, so Fluent folks, forgive me if this is not true. I read Roe's original paper but constantly find myself lost, mainly due to my poor command of mathematics. Regards, Shenren Last edited by Shenren_CN; October 11, 2012 at 21:51.

October 11, 2012, 21:50
#2
Member

Shenren Xu
Join Date: Jan 2011
Location: London, U.K.
Posts: 63
Rep Power: 8
Found two references online from people that I think know what they are talking about. Hope this can be of help to someone asking the same question.

1) From him (http://www.ita.uni-heidelberg.de/~dullemond/index.shtml)

2) From him (http://www.hiroakinishikawa.com/)

http://www.cfdnotes.com/cfdnotes_roe...d_density.html

Quote:
 Originally Posted by Shenren_CN Dear colleagues, Could anyone please explain to me or point me to some reference on why Roe's averages are so important in Roe flux scheme. A simple question I have is what happens if I simply take the arithmetic average or geometric average of the left and right state, instead of strictly follow the Roe's average formulation. In fact, I remember I read in Fluent's manual that the Roe's scheme implemented therein uses simple arithmetic average, don't know sure. Couldn't find the link now, so Fluent folks, forgive me if this is not true. I read Roe's original paper but constantly find myself lost, mainly due to my poor command of mathematics. Regards, Shenren

October 12, 2012, 03:26
#3
Member

Francesco Capuano
Join Date: May 2010
Posts: 81
Rep Power: 9
Quote:
 Originally Posted by Shenren_CN Dear colleagues, Could anyone please explain to me or point me to some reference on why Roe's averages are so important in Roe flux scheme. A simple question I have is what happens if I simply take the arithmetic average or geometric average of the left and right state, instead of strictly follow the Roe's average formulation. In fact, I remember I read in Fluent's manual that the Roe's scheme implemented therein uses simple arithmetic average, don't know why. Couldn't find the link now, so Fluent folks, forgive me if this is not true. I read Roe's original paper but constantly find myself lost, mainly due to my poor command of mathematics. Regards, Shenren
You can find a very nice answer to your question in the book "Finite-volume methods for hyperbolic problems", by R.J. Leveque, pages 315-320.

In a few words, if one replaces the nonlinear problem dq/dt + df(q)/dx with a linearized Riemann problem at each interface, dq/dt + A*dq/dx, then a problem arises in defining the Jacobian A. A natural choice could be

A = f'(qavg), where f' = df/dq

One could simply use qavg = 0.5(qr + ql), averaging right and left states (as Fluent does). For some problems, however (e.g. Euler equations, shallow water equations, ...), a special average value exists for which a number of "nice" properties are satisfied - and that's why Roe's average is important.

In some way, the idea is more or less the same of using the mean value theorem between right and left states - although I'm not sure about the mathematical formalism for this.

October 12, 2012, 07:10
#4
Member

Shenren Xu
Join Date: Jan 2011
Location: London, U.K.
Posts: 63
Rep Power: 8
Ciao Francesco,

Could you please elaborate a little bit more on the 'nice' properties of Roe's averages and why those properties are important to a good simulation? You see, my question is exactly why the Roe's averages are superior than using the arithmetic average, which as you confirmed, is what Fluent does.

I'll read the book you recommended. Thank you!

Shenren

Quote:
 Originally Posted by francesco_capuano You can find a very nice answer to your question in the book "Finite-volume methods for hyperbolic problems", by R.J. Leveque, pages 315-320. In a few words, if one replaces the nonlinear problem dq/dt + df(q)/dx with a linearized Riemann problem at each interface, dq/dt + A*dq/dx, then a problem arises in defining the Jacobian A. A natural choice could be A = f'(qavg), where f' = df/dq One could simply use qavg = 0.5(qr + ql), averaging right and left states (as Fluent does). For some problems, however (e.g. Euler equations, shallow water equations, ...), a special average value exists for which a number of "nice" properties are satisfied - and that's why Roe's average is important. In some way, the idea is more or less the same of using the mean value theorem between right and left states - although I'm not sure about the mathematical formalism for this.

October 12, 2012, 08:52
#5
Member

Francesco Capuano
Join Date: May 2010
Posts: 81
Rep Power: 9
Quote:
 Originally Posted by Shenren_CN Ciao Francesco, Could you please elaborate a little bit more on the 'nice' properties of Roe's averages and why those properties are important to a good simulation? You see, my question is exactly why the Roe's averages are superior than using the arithmetic average, which as you confirmed, is what Fluent does. I'll read the book you recommended. Thank you! Shenren
Hi Shenren,

unfortunately a comprehensive answer to your question needs a detailed mathematical formulation, which you can find in the above-mentioned book. In simple words, use of arithmetic average would result in a spurious behaviour near shocks, which on the contrary would be well-resolved using Roe's average - see Figure 15.2 from Leveque's book. This is a consequence of some mathematical constraints which are violated using a simple average.

Unfortunately, for practical problems a Roe linearization is not available and many solvers (e.g. Fluent) use arithmetic average (which, however, works well in regions where the flow is smooth).

Hope this helps,
Francesco

October 12, 2012, 09:04
#6
Member

Shenren Xu
Join Date: Jan 2011
Location: London, U.K.
Posts: 63
Rep Power: 8
Hi Francesco,

1) I'm reading the book you recommended and as you explained, the superiority of Roe averages indeed is much more evident when there's large gradient in the flow, such as shocks. No wonder Roe scheme is known for shock capturing.

2) I thought Fluent uses a simple average only because of its simplicity. In what circumstance would Roe's linearisation be not available? Could you give one or two simple examples? In that case, what shall we do for shock-capturing in those cases then?

Regards,
Shenren

Quote:
 Originally Posted by francesco_capuano Hi Shenren, unfortunately a comprehensive answer to your question needs a detailed mathematical formulation, which you can find in the above-mentioned book. In simple words, use of arithmetic average would result in a spurious behaviour near shocks, which on the contrary would be well-resolved using Roe's average - see Figure 15.2 from Leveque's book. This is a consequence of some mathematical constraints which are violated using a simple average. Unfortunately, for practical problems a Roe linearization is not available and many solvers (e.g. Fluent) use arithmetic average (which, however, works well in regions where the flow is smooth). Hope this helps, Francesco

October 12, 2012, 14:02
#7
Member

Francesco Capuano
Join Date: May 2010
Posts: 81
Rep Power: 9
Quote:
 Originally Posted by Shenren_CN 2) I thought Fluent uses a simple average only because of its simplicity. In what circumstance would Roe's linearisation be not available? Could you give one or two simple examples?
I agree that Fluent uses simple average due to its simplicity, also because the exact form of Roe's linearization depends on the specific nonlinear problem to be solved - and Fluent can solve multiple problems. Actually I'm not really sure about cases in which it is not formally possible to derive a Roe-like average: I was thinking about real-gas mixtures or reacting flows, but - at first glance - it seems from literature that some work has been done in this direction too... maybe other people from the forum can give us a clue.

Quote:
 Originally Posted by Shenren_CN In that case, what shall we do for shock-capturing in those cases then?
Roe's linearization is not the only way to achieve satisfactory shock-capturing. Many other numerical schemes (e.g., flux-vector splitting schemes, central schemes with ad-hoc dissipation, as well as higher-order schemes, etc.) are available which can provide good results - you can find an enormous amount of literature about that, depending on your specific application.

As a general consideration, however, I think you should ask yourself about the real importance of numerical accuracy in your activity: if you are doing "engineering" work, and you are concerned with global properties of the flow (let's say, drag coefficient, heat flux, etc.), then I wouldn't be too worried about the details of the flux scheme - just try to achieve a solution which is grid- and scheme-independent and very good confidence would be placed in your results. On the other hand, a very different situation arises if you are doing scientific work, then numerical accuracy becomes a primary issue... but that is just my two cents

Regards,
Francesco

 October 14, 2012, 09:44 #8 Member     Serge A. Suchkov Join Date: Oct 2011 Location: Moscow, Russia Posts: 74 Blog Entries: 5 Rep Power: 7 Keep in mind that Roe averaging dramatically slower than simple linear averaging __________________ OpenHyperFLOW2D Project

October 14, 2012, 10:24
#9
Member

Shenren Xu
Join Date: Jan 2011
Location: London, U.K.
Posts: 63
Rep Power: 8
Hi SergeAS,

I admit that Roe averaging is more costly then linear averaging.
But is it really dramatically slower? I thought the most costly part
is the wave amplitude \times the wave speed, i.e., |A|\cdot \delta_w.
But I may be wrong, it's just that your argument is not apparent to me.

Regards,
Shenren

Quote:
 Originally Posted by SergeAS Keep in mind that Roe averaging dramatically slower than simple linear averaging

 April 19, 2013, 14:26 #10 Senior Member     Ehsan Join Date: Oct 2012 Location: Iran Posts: 2,210 Rep Power: 19 Hi Is rhoPimpleFoam in OpenFOAM appropriate to capturing of shock wave? does Roe term is div(phi,U)?

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is OffTrackbacks are On Pingbacks are On Refbacks are On Forum Rules

 Similar Threads Thread Thread Starter Forum Replies Last Post ares Main CFD Forum 10 May 7, 2010 04:47 zouchu Main CFD Forum 3 August 10, 2000 16:46 Jian Xia Main CFD Forum 7 August 9, 2000 01:18 Mohammad Kermani Main CFD Forum 6 December 29, 1999 12:11 Mohammad Kermani Main CFD Forum 5 December 20, 1999 16:44

All times are GMT -4. The time now is 05:13.