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elankov November 2, 1998 05:19

Finite Difference Vs. Finite Volume

I m a beginer in CFD. From the five commercially available softwares for CFD which i know four of it uses finite volume methods such as PHOENICS, FLUENT, STAR-CD and FLOW3D. Can anyone explain why finite difference method is not prefered?


John C. Chien November 2, 1998 13:44

Re: Finite Difference Vs. Finite Volume
It is a good question. In uniform mesh, they are identical in many cases. In a way, the finite volume approach gives an impression that the overall mass, momentum etc. are conserved over a volume. It is also related to how people formulate the equations, that is the form of the equation before discretization. In the divergence form, the integration may seem to be easier to do. Anyway, I think it is easier to hide the assumptions under the finite volume method without mathematical justification. In the finite difference approach, you have to cover every terms generated from the coordinate transformation for general 3-D problems. In many cases, the evaluation of these terms are sensitive to the quality of the mesh used. As a result, a good finite difference solution is always more accurate than the finite volume solution because you have to pay attention to many more detail areas. The other reason is the influence from the finite element method which is more flexible for complex geometry. So, if you are looking for more accurate solutions, you may want to use finite difference methods. Otherwise, finite volume method will give you a solution, which may not be accurate enough, and you will be forced to refine the mesh ( volume or cells ) on and on . ( but at least that is the users problem. )

Duane Baker November 2, 1998 15:44

Re: Finite Difference Vs. Finite Volume
I think that we sould not mislead the poor beginner here by saying anything like: "As a result, a good finite difference solution is always more accurate than the finite volume solution because you have to pay attention to many more detail areas." The fact is that if the same level of attention is paid for both of the methods then the roughly the same accuracy is obtained.

First of all it is useful to look at the "method of weighted residuals" as the Mother of all methods. We can then look at all of the methods as decendents and relate the differences in a logical fashon rather than broad, unfounded, and often inacurate statements. In this manner the discrete equations for each method are obtained from the differential equations and the methods only differ in the weight function used.

1.FD is seen to be a weighted residual method with the Dirac Delta as weight function at the node point and zero everywhere else.

2.FV is seen to be a uniform weight function over the cell and zero everywhere else.

3. FEM is seen to be the weight function is the same as the shape function (Galerkin or Bubnov-Galerkin) or some variant of the shape functions as in Petrov-Galerkin.

Now, anybody who says that a technique using nodal Dirac-Delta is ALWAYS MORE ACCURATE than the other methods had better rethink their basic understanding of the methods.

For the poor beginner here are the fundamental advantages of the FV method:

1. Engineers like it because the Integral conservation equations (unity weight function over discrete volume) when expressed in conservation form (divergence of fluxes) can be convert the volume integrals to surface integrals using Gauss Divergence Theorem. This is a direct extension of the control volume analysis that Engineers are used to in Thermo, Heat Transfer, etc and can then be easily interpreted and troubleshooted.

2. If care is taken to obtain a conservative discrete FV method, exact global conservation is ensured for all grids, not only in the limit of grid refinement. Most industrial geometries and and boundary conditions are such that the grid-independant asymptote is a long ways away with practical grid size (remember 100 X 100 X 100 is a Million nodes)! It is hard to convince a turbine manufacturer that you can improve the efficiency from 95% using CFD to 96% when you have a mass or momentum loss of 5% in your simulation! This is not to neglect the importance of grid refinement studies but just being practial! And I will bet that the coarse grid strongly conservative FV solution is a better design tool than the FD. It is obvious from the vendors that the preferend method is FV. You can also add to that list CFX-TASCflow as a strongly conservative control volume based finite element method....probably the most accurate commercial code available.


John C. Chien November 2, 1998 18:19

Re: Finite Difference Vs. Finite Volume
Yes, you are right about it. But the seemingly good results with coarse mesh used can also be misleading. The other point you are trying to make is the so-call "weighted residual " procedure. I think, you can fit the three different methods into one formulation, but in reality, FD method is normally derived from the partial differential equations and no additional assumptions are made on the solution behavior. On the other hand, whether the "weighted residual " procedure of FE will produce real solution to the Navier-Stokes equations is, I think, still an open ended question. In many cases, the solution is of exponential funtion type, therefore, by assuming ahead of time that the solution distribution is either linear or parabolic is simply an approximation at best. Whether it is related to the real solution or not is questionable. For real 3-D problems, it is really hard to address the code accuracy without solving a real problem. Only the solution accuracy is meanful in this case. For FV method, since the overall conservation is satisfied, the flow variable will be off. The accuracy of flow variable has to be sacrificed.

Andy Thompson November 3, 1998 10:10

Re: Finite Difference Vs. Finite Volume
Duane A very well thought out and logically justified argument, leading to a sensible conclusion - right up to the point where you CLAIM CFX-TASCflow is "probably the most accurate commercial code available"!! If you are going to use this DISCUSSION FORUM for blatant commercialism, then maybe another well thought out and logically justifiable argument for your conclusions would help?!?!?!

John C. Chien November 3, 1998 10:53

Re: Finite Difference Vs. Finite Volume
I think, it is a common sense that FD is more accurate than FV based on my experience. Mybe a John Chien's conjesture is more appropriate for this occasion :" On a smooth, non-uniform 3-D curvilinear mesh, FD solution is always more accurate than FV solution."Nov.3,1998. A concept which is based on the control volume approach (1-D) is not going to be more accurate than traditional finite difference approach. I have never mentioned any commercial code in the Internet by name, therefore, I am not going to touch the issue of which code is using what method or which code is more accurate. As I said before, it is the code developer's responsibility to provide benchmark test results to the users. I would like to see the benchmark results published here. It's not going to take another 300 years to solve John Chien' conjecture. I am not claiming that I am always right. Maybe 300 years from now, everyone will be using the unified FE method. And everyone over 77 will have to live in the space station. And everyone will ask: " John Chien's conjesture ? What ?." The Internet is so big that my feeling simply don't exist at all. After all, a common sense is a common sense. And John Chien's conjesture shouldn't take 300 years to solve.

Andy Thompson November 3, 1998 11:11

Re: Finite Difference Vs. Finite Volume

I am not in a position (intelectually) to argue with you over your conjecture!! I have read many of your inputs to this forum and they have always been informative and unbiased - attributes which I am sure are appreciated by contributors from both research and commercial areas. I do agree with you when you say it is the code developers responsibility to provide benchmark results, something which is all too often forgotten.

Keep up the interesting contributions!!

Philip Zwart November 3, 1998 11:58

Re: Finite Difference Vs. Finite Volume
Well, let me throw my hat into the fray and try to answer your question about why the finite volume method is preferred by commercial codes. John has said that FD is usually more accurate, and Duane has said perhaps not. Resolving this question is highly complex because, within both the FD and FV methodologies there are many choices which must be made for the discretizations of the various terms, and these choices are probably more important than choosing FD or FV.

Apart from what has been discussed, I think it is worth mentioning a few points which have not yet come up. First, for hyperbolic systems, the differential equations do not hold at discontinuities, whereas the integral conservation laws do... this is a point in favour of FV.

Second, FV is more easily extended to unstructured meshes, but I haven't seen any serious effort at FD on unstructured meshes... perhaps someone can correct me on this. The trend in commercial codes is definitely toward unstructured technology capable of handling complex 3D geometries.

Third, accuracy in the limit of fine meshes is not the only factor commercial codes must consider. More important from their perspective is robustness for a wide variety of general problems and meshes. Here, FV appears to win out over FD, because it enforces strong conservation at all times. Why is this important? Well, I'm not sure I completely know the answer to that, but one important factor is that the algebraic equations that come out of the discretization have nice properties (eg. diagonal dominance). Mass conservation is particularly important... see Patankar's book for more discussion of this.

A related issue is that commercial codes want their solutions to be physically reasonable even on coarse meshes. This is a different issue than accuracy in the limit of very fine meshes.

I hope this is a helpful addition to the discussion.


John C. Chien November 3, 1998 14:08

Re: Finite Difference Vs. Finite Volume
I agree with you on everything you said. The global conservation and the complex 3-d geometry are two main driving force behind the FV method. But one has to realize that turbulent flows are the main focus today. In this case, the velocity gradient field is more important than the mass and momentum conservation. In the turbine cooling area, cooling holes are used everywhere. People have been using just one cell to represent the jet in the FV calculation. There's no problem in getting a solution. But in reality, even with one hundred thousand cells, the jet-in-a-cross-flow problem is still far from solved. Last year, when I was calculating 3-D combustor flow using an in-house FV code, I had only one cell to cover the jet port. I think, I was just kidding myself. Being able to present a solution does not mean that the solution is right. If people of CFD in early days were able to say that it did not make any sense to represent the jet with just one cell, the progress in this area would be far more advanced today. The important area of loss prediction in turbomachinery is a strong function of the pressure and velocity gradients, not a direction function of mass conservation. The secondary flow loss ( requires accurate 3-D flow field), the tip-clearance loss ( flow through very small gap ), the wake loss ( accurate description of the wake generation and development ), the disk cavity heat transfer, etc...all depends on the accurate 3-D flow field gradients to provide the loss calculation. Just overall correct 3-D flow field is not going to do the job. The same is true for the pump design, where the CFD is used to improve the efficiency. The ability to compute the 3-D flow field is one thing, the ability to improve the efficiency is another story. ( you may want to ask Professor Patankar or Dr. Rodi about how many cells are required to predict accurately the jet-in-a-cross-flow problem. So, what I am trying to say is that the FV code I used last year should gave me an error message saying : John, you can't do that, you need to define the jet port with a minimum of ..... cells in order to get a reasonable answer.) By the way, I support anyone working in CFD field; good, bad or ugly.

Duane Baker November 3, 1998 15:59

Re: Finite Difference Vs. Finite Volume

firstly, thank you for your comments on the majority of my posting. Now, for the commercialism accusations:

commercial - Having financial gain as an object. blatant - obvious or obrusive

......from Funk and Wagnalls standard desk dictionary

I neither work for nor profit in any way from AEA Advanced Scientific Computing!

WHO DO YOU WORK FOR? (from your email address it is BLATANTLY OBVIOUS)


I work for the Alberta Research Council and have used several commercial codes, including CFX-TASCflow, Fluent, PHOENICS, etc. I have also studied and written my own CFD codes and I am fairly well versed in the literature on CFD developments. As a result of this, I have come to the independant conclusion that CFX-TASCflow IS PROBABLY THE MOST ACCURATE commercial code available from the general purpose CFD codes which I have used. My comments also reflect the opinions of other users which I have gathered through various discussions, particulary discussions with some German Enginners (Probably the best Engineers in the world!...this is meerly my opinion and not a request for all dogmatic suppressors to now post their accusations). Further supporting evidience is the fact that two years ago AEA purchased ASC when they already had a CFD code....why do you think that was???

On a technical basis, the coupled momentum and continuity solver approach of CFX-TASCflow combined with the algebraic multi-grid linear slover is the fastest and most robust solver that I have ever used. What does Fluent (or any of the other codes for that matter) offer in this respect? The last time I used Fluent, the number of interequation relaxation parameters which required tuning resulted in more of a black-art than a scientific approach to getting a converged solution. As for discretization, the Linear Profile Skew Upstream Differencing Scheme (LP-SUDS) with Physical Advection Correction (PAC) provides the most accurate and robust second-order treatment of advection that I have ever used. The Mass Weighted SUDS provides an extremly accurate first order scheme for those tough industrial problems on coarse grids that require a very robust scheme. What does Fluent offer in these respects Quick and UDS?? If I recall, Fluent was still using the staggered grid in the early 90's when ASC has a colocated non-orthogonal boundary fitted grid with the control-volume finite element method since about 1986! That has of course been rectified by implementing the Rhie and Chow technique, somewhat like following what ASC had done a long time before!

In my posting, I was meerly adding my experience and the CFX-TASCflow code to the list of 4 which was originally posted so that the beginner can broaden his scope of codes. I believe this is consistient with the purpose of this DISCUSSION FORUM and would encourage you to to be CONSTRUCTIVE by adding some valuable discussion of the details of what your company can provide (some of my knowledge may be out of date)!

Duane Baker P.Eng., B.Sc.(Mec E), M.A.Sc.(Chem E), M.A.Sc.(Mech E) CFD Research Engineer Alberta Research Council Email: baker@arc.ab.caane Baker

John C. Chien November 3, 1998 18:09

Re: Finite Difference Vs. Finite Volume
I am very happy to hear that there is a satisfied customer of a commercial CFD code. The goal of a commercial product is not to kill other similar products but to satisfy the need of a customer. The performance is secondary . This is very true in car business. In TV ads of new cars, you rarely see two cars racing against each other. Because it does not make sense to do that. And no one will be happy when there is only one commercial code to choose from. I think every code developer should receive a medal because they are the ones who took the risk to develop the codes.

James Bond November 4, 1998 06:11

Re: Finite Difference Vs. Finite Volume
'Further supporting evidience is the fact that two years ago AEA purchased ASC when they already had a CFD code....why do you think that was??? '

This is easy as most in the industry know. CFX 5 had just been released ... and bombed. Yep it was being eaten alive by STAR CD and FLUENT/UNS. To stay with the game AEAT had to do something. That something was the purchase of ASC.

Andy Thompson November 4, 1998 06:56

Re: Finite Difference Vs. Finite Volume
Duane, With apologies for my earlier tone, and in the most conciliatory manner I can muster - I was obviously unaware of your allegiance (i.e. NOt to a commercial CFD code vendor), and I over reacted to what I maintain was an unsubstantiated statement in your first posting. If you had stated that your conclusions had come from your (obviously impressive) experience of the various commercial codes available, I am (almost) certain I would have reacted differently. Anyway - apologies again.

John C. Chien November 4, 1998 10:40

Re: Finite Difference Vs. Finite Volume
When a CFD code becomes a commercial code , it becomes a black box by definition in order to survive in the marketplace. When a black box meets a black box they can't see each other, they have to wear colorful masks. You can't ask CoKe to reveal its secret formula. The same is true that you can't ask an Olympic figure skater to post nude when competing in the game. But in order to see the underside of a black box, sometimes the user must ask questions to provoke discussions with negative questions. I don't feel bad at all because it is part of my intention to help bring out feeling from inside the black box. We as users also have to learn how to survive through the next century. By the way,in the traditional Chinese opera, female characters are always male actors. What's inside the black box really is not important at all. Audience's satisfaction perhaps is the most important issue.? Or is it ? How do you know he really understand the Chinese opera ? But there is no question that the use of FV in commercial code is today's FASHION. It is an interesting subject of discussions.

Andy Thompson November 4, 1998 11:30

Has the subject changed??
Considering my position within the CFD industry is in a Consultancy capacity, my experience lies in application of the techniques to industrial problems, often for engineers with little/no CFD knowledge. I have only a limited knowledge of the "inside of the black box" - just enough to be able to choose the most appropriate models/techniques for the particular problem in hand. This comes as much from experience as from knowing HOW the model works! As far as the audience (industrial client) is concerned, there is often no need for them to know what we do to achieve a solution (unless they ask for specifics). Their main aim is to solve an engineering problem, CFD is just another tool being used to help them find the solution. As with all tools, the result depends as much on the expertise of the user as the quality of the tool. Obviously the tool is being continually refined and updated, and the methods of use are constantly under review, but surely the fact that engineers are being provided with more insight and better understanding is more important than giving them an answer to 5 decimal places!?!?!

John C. Chien November 4, 1998 12:54

Re: Has the subject changed??
You do bring up an interesting point, maybe the FV formulation in a commercial code provides a perfect match for the average engineers who are mainly concern about the ability to obtain a solution to their problems. The average engineers are not equipped to deal with the complex coordinates transformation factors associated with FD mesh. It does make sense to think in this way. The conclusion is : FV is more user-friendly for average engineers, and FD is more complex for advanced professional. So, the FV used in the commercial code is purely a business consideration. And if the advanced professionals are not satisfied with FV approach, they are free to write their own code in FD. ( I think they have been doing this for a long time already.) By the way, the subject has not changed. I am just wondering why the FE fashion of 70's simply evaporated in late 90's.

Andy Thompson November 4, 1998 13:09

Back to FD vs. FV
I really wish I could answer the question about the change in fashion - maybe it is something to do with ease of implementation, or maybe it is just that all us AVERAGE ENGINEERS can't deal with the complexities which all you ADVANCED PROFESSIONALS eat up for breakfast. I can state however, that some of the problems that are solved on a day to day basis by "average engineers" using commercial CFD codes would make many an "advanced professional" sit up and take notice!!

John C. Chien November 4, 1998 14:03

Re: Back to FD vs. FV
I am quite amazed by today's commercial CFD codes' capabilities to handle complex geometry. In this area, you got an A+. But if you remove the supporting staff and service from the day-to-day CFD activities, the average engineers would be left in the cold alone. So it is important to have experienced CFD engineers to provide the service to the average engineers to solve their daily problems. And if an average engineer received his (or her ) training at school in FD form ( numerical methods, partial differential equations , etc..), he is likely to ask questions about the commercial use of the FV methods. Unless the exact FV method used in the code is explained in detail,the question of " FD vs FV " will remain largely un-answered. For this reason, benchmark test cases and samples are the only way to guide you in the selection of commercial CFD codes. In this area, experienced CFD engineers will play a key role in saving users ( and companies) a great deal of money and time.( until a smart FV code can be developed to warn the users automatically in regard to the solution accuracy and coarse mesh issues.) I think, the accuracy issue and the coarse mesh solution problem are still important to smart users .

krishnakumar November 4, 1998 16:08

Re: Finite Difference Vs. Finite Volume
I am a beginner in CFD and I was surprised to find that I was able to follow quite a lot of ur discussion. But I still do have some doubts. From ur last message (chein), I think ur trying to say that FD is better than FV for solving turbulent flows. How are FD schemes better for turbulent flows? And as for the minimum discretisation question, I think FD and FV stand equal. Neither can u directly decide on the number of cells(FV), nor for the number of nodes(FD) for getting reasonable answers.Please correct me if I am wrong. I also want to point out something interesting. I am doing a project on a least squares scheme at present. It is actually an FD scheme, based on the least squares method of reducing the differntial error to find the spatial derivatives(flux derivatives). A very robust scheme, very similar to this one, called LSKUM(least squares kinetic upwind method) can handle very complex geometries indeed.( I cant remember the exact ref for this now, but I will find it out and post it here as soon as possible.) Infact, this scheme works on a cloud of points and can work out the solution given just the connectivity relation between these points. The scheme (using differnce spliting rather than vector splitting) I am involved with is the same as FV(mathematically) in a uniform mesh, and the diff comes in a non-uniform mesh. We haven`t yet found whether this diff is for the better or for the worse, but the intresting point here is the similarity. I think future schemes will use the techniques of FV,FD,FE,Lsq(least squares or other statistical estimators) and it will be pretty tough to characterise schemes as purely FD or FV. Please correct me if I am wrong here.

(reference to LSKUM will soon follow..)

Krishnakumar November 5, 1998 04:07

Ref for LSKUM
Reference for LSKUM :

AIAA(1995) - A.K.ghosh & S.M.deshpande - AIAA-95-1735.

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