Dear All, I'd like to post the following questions:

(1) Numerically, does a higher-order approximation always yield more accurate results than the relevant lower-order one ?

(2) Mathematically, the unstructured FVM is regarded as a special case of the FEM in terms of the lower-order weighting functions. But why is the former much more popular than the latter ?

thx in advance.

Tony

Hi!Hony:

(1) Using a higher-order approximation, sometimes, yields more accurate results, but only deals with low Reynolds number in FVM, in my experience.

(2) FVM is not so accurate as FEM, but needs a relative small computing resources(memory, CPU time). Specially, for large scale problems, FVM is available to run on a PC, but FEM employing directing solutions is not.

Best regard, Huashu

FEM is not so popular than FVM because FEM still has problems in treating convection terms especially for high speed (high Re) flow where the convection is dominant.

(1) In order to get a precise answer you would have to expand what you mean by accuracy. Some numerical methods are good at simulating certain physical properties and poor at others. A woolly (but usable) answer to the question is that it depends on how well the grid has resolved the flowfield. Traditional high order methods require the flowfield to be smooth and, if it is, they generally give very "accurate" answers. However, if the flowfield has not been well resolved they usually predict excessive oscillations which can look like nonsense. Inadequately resolved flowfields using low order methods generally look much more sensible (which can be a problem in itself!).

(2) I would have liked the word "can be" in there. Another woolly and very incomplete answer. A normal Galerkin formulation which would arise naturally from following a typical structural derivation does not work for high Reynolds flows and would be inefficient to solve anyway. To fix things one typically adopts non Galerkin methods and mass lumping ending up with something that is pretty much a traditional FVM. In the process one tends to lose the wide range of element types which work (one of the major attractions of the FEM for structural analysis). So why label these schemes FVM instead of FEM? Probably because a long time ago they were first derived by engineers using a direct finite volume/finite difference approach on structured grids rather than a FEM approach. Their popularity almost certainly stems from the fact that they are simple and have been widely demonstrated to work. Had these engineers originally developed and promoted the schemes under the FE label I am sure there would be a lot less FVM schemes around today.

Hi Tony,

1) In most practical cases a lower order scheme will result in a more accurate solution than a higher order one on a very coarse grid and vice versa as the grid is refined. This explains why first order upwinding was popular until the 1980's when people began to refine their grids (not to mention the iterative solver convergence problems that arise if diagonal dominance is lost as often happens with higher order methods).

2) Strictly speaking, all of the FD, FV , and FEM can be shown to be cases of the method of weighted residuals. The FV method has a weighting function of 1, Galerkin has the basis functions themselves, and FD is a Dirac delta.

Just like the history in solid mechanics methods, the FEM has taken longer to develop. Early works in the solid mechanics were primarlily FD until the mid to late 1960's when people like Zienkiewicz moved in with the FEM. If the solid mechanics work is any kind of a foreshadow, FEM may take over some day in CFD! The basic issue is on the level of understanding or comfort of the practicioner. The FEM requires considerably higher mathematical understanding while FVM relies upon more intuitive principles (from an engineering point of view)!

What do you mean by "lower order weighting functions"??? I have only ever seen 1 as THE weighting function on the exact integral over the volume!

There was an extensive discussion on this topic in the last 4 months or so......maybe you can still retrieve it???

Regards......................................Duane

hello,

(1) Higher order methods are always accurate than their Lower order counterpart. there is no doubt about it.

(2) FVM is accurate than FEM b'cos FVM adopts higer order discretization whereas FEM is of lower order, in terms of accuracy. see the basic difference between FVM and FEM is FVM represent the governing equation in the conservsative form as vloume integral whereas in FEM one used Taylor series approximation of the derivatives(of unknowns). there is quite a good difference in term of discretization accuracy between FVM and FEM and it's not the history, what you said.

(3) see the basic difference between FDM and FEM is finite difference methods use only uniform grid and FEM can use non-uniform grids. that's why they are more commonly used in case of analysising complex geomentries. but if one talks in term of accuracy definetly finite difference methods are lot more accurate then finite elements. see it's not that one can't solve complex geomentries in finite difference or by finite volume, it can be done, if one works in transformed plane then in physical plane, where a non-unifom grid in physical plane can be transformed into uniform grid in transformed plane.

(4) any day FVM can use a non-uniform and unstructed grid like what FEM uses. that's where the superiority of FVM comes into play. still the accuracy can be increased if one used Flux Vector splitting methods with QUICK and MUSCL type of schemes. the stability is also ensured by varouis methods like Flux limiting, Slope limiting, etc...

regards,

Sridar D. if you want please refer to,

Hirsch C., Numerical Comp. of Internal and External Flow, vol1, John Wiely & Sons.

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(1) Higher order methods are always accurate than their Lower order counterpart. there is no doubt about it.

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I am afraid there is. A high order method will only be more accurate than a low order method if the truncation term is smaller. If the solution is smooth (small higher order derivatives) this will happen but one cannot make the statement otherwise (unresolved).

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(2) FVM is accurate than FEM b'cos FVM adopts higer order discretization whereas FEM is of lower order, in terms of accuracy. see the basic difference between FVM and FEM is FVM represent the governing equation in the conservsative form as vloume integral whereas in FEM one used Taylor series approximation of the derivatives(of unknowns). there is quite a good difference in term of discretization accuracy between FVM and FEM and it's not the history, what you said.

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The order of accuracy depends on the order of discretization scheme used with the FV, FD or FE and not the approach. It is rare to use a Taylor Series Expansion to derive a finite element formulation. There is little difference between the simplest usable forms of FEM and FVM formulations - the most noticable difference is usually the diffusion stencil.

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(3) see the basic difference between FDM and FEM is finite difference methods use only uniform grid and FEM can use non-uniform grids. that's why they are more commonly used in case of analysising complex geomentries. but if one talks in term of accuracy definetly finite difference methods are lot more accurate then finite elements. see it's not that one can't solve complex geomentries in finite difference or by finite volume, it can be done, if one works in transformed plane then in physical plane, where a non-unifom grid in physical plane can be transformed into uniform grid in transformed plane.

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As with the order of accuracy, the type of grid (structured or unstructured) is not prescribed by the FD or FE approach. Similarly, the use of curvilinear coordinates to evaluare spatial derivatives can be used by all approaches.

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(4) any day FVM can use a non-uniform and unstructed grid like what FEM uses. that's where the superiority of FVM comes into play. still the accuracy can be increased if one used Flux Vector splitting methods with QUICK and MUSCL type of schemes. the stability is also ensured by varouis methods like Flux limiting, Slope limiting, etc...

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?

(1). The questions asked really have no meaning at all. (2). I guess, you are talking about the truncation errors in the Taylor series expansion of a derivative term. The existence of derivatives, or higher order derivatives depend on the local solution behavior. It can be a constant value, or solution with first order derivative. When the local solution is a constant, there is no first order derivative and higher order derivatives. When the local solution is discontinuous, the derivatives don't exist at all. (3). Finite-difference, finite-volume and finite-element methods are derived through different processes. They are not related. If they are related, then you can derive the finite-volume method from the finite-difference method. (4). The finite-difference method requires the use of coordinate systems and Taylor series expansion of the derivatives in the equations. For arbitrary geometry, a general coordinate transfomation is required to transform the equations. Equations are normally solved in the transformed coordinate system. (5). The finite-volume method does not deal with the governing equations in the differential form. The integral form of the governing equations is used instead. This is different from the finite-difference formulation, which deal with the differential form directly. The approximation is applied to the integral form of the equations. (6). The finite-volume method is completely different from the above two methods. First, it requires the explicit assumption of the solution form, whether it is linear, or second-order,etc. Then a minimization process is applied to the equation using variational principle, weighted-resudual methods,etc. to derive the final algebraic equations. It is important to know that the use of the finite-element method requires the assumption of the form of the solution. If a linear form is used, the solution is diamond-face like solution. You can't draw a smooth curve through the nodal points when the solution is obtained. (7).As for the popularity of the finite-volume method, it is really hard to say because the source code of a commercial CFD code is always closely guarded. Unless you have seen all of the source codes, no one really know exactly what is used. That is why I said these questions have no meaning at all.

(6). The finite-volume method ....should be (6). The finite-element method ....

Hi Duane,

Thx, you just answered my questions. And thx to all.

Tony

hello,

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I am afraid there is. A high order method will only be more accurate than a low order method if the truncation term is smaller. If the solution is smooth (small higher order derivatives) this will happen but one cannot make the statement otherwise (unresolved).

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Please take a look at following paper

T. K. Sengupta & Kapil Gupta, Tailoring Higher Order Methods for DNS and LES, 3rd Asian Computational Fluid Dynamics Conference, vol1,93-103,Bangalore,India.

let me give a brief of what is their. if you want to find the accuracy and stability of any method you have, write your method in spectral form (i.e. take fourier transform of it) compare or plot the ratio of spectral equivalent of the discretised eqn. and the exact spectral representation of the governing equation for all possible wave no. then if you compare among the various methods, i think you will clear by youself.

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As with the order of accuracy, the type of grid (structured or unstructured) is not prescribed by the FD or FE approach. Similarly, the use of curvilinear coordinates to evaluare spatial derivatives can be used by all approaches.

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yaa that's true, curvilinear coordinate representation of spatial term is common for any approach. if that is the fact then one can prefer to go for a higer order finite difference approach then FE approach. am i right!!. if you take look at the same paper, i mensioned above, you will find a higher order compact difference scheme representation is almost equivalent to the spectral representation.

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please refer the book i mensioned in last response. in addition you can refer to

D. M. Causon et al, On application of High resolution Shock Capturing Methods to Unsteady Flows, Numerical Methods for Wave Propagation, 145-171, Kluwer Acad. Press, Netherlands.

regards,

Sridar D.

Hi John,

not to get into too much of a battle here, but in regards to the statement:

Finite-difference, finite-volume and finite-element methods are derived through different processes. They are not related. If they are related, then you can derive the finite-volume method from the finite-difference method.

1. They are TYPICALLY derrived by different processes!

2. They can and it may be useful to derive all from a common method: the method of weighted residuals. See Fletcher Vol I, chapter 5.

Regards, Duane

Hi

You said that

(2) FVM is accurate than FEM b'cos FVM adopts higer order discretization whereas FEM is of lower order, in terms of accuracy. see the basic difference between FVM and FEM is FVM represent the governing equation in the conservsative form as vloume integral whereas in FEM one used Taylor series approximation of the derivatives(of unknowns). there is quite a good difference in term of discretization accuracy between FVM and FEM and it's not the history, what you said.

Can you be more explicit on that? I have been working on finite element method (FEM) for 4 years now and I have never seen Taylor series approximations of the derivatives. For me FEM is just a Galerkin (or Petrov-Galerkin) approximation of the weak form of the evolution equations with piecewise polynomials (usually piecewise linear in high Re CFD but in principle higher order polynomials can be used) as test functions.

One main advantage of the FEM for me is its ability to deal with boundary conditions (Dirichlet conds as well as Neumann conds)

FEM is proved to be optimal for elliptic problems (Stokes flows,...) but what concerns hyperbolic problems it 's an other business

Gary

(1). Many many years ago, I had a book on weighted residuals published by Academic Press. I remembered that there was a simple example which illustrated the equivalence of FEM and FD under certain conditions. I think, it is possible to obtain the same final algebraic equation from different methods under certain conditions. But, in general, they are not related. That is why there are three unique names. That condition mentioned in the book was something like " under uniform rectangular grid". (2). For elliptic equations, (heat conduction equation) the approach taken by FE is a valid approach. For other more complicated systems of equations, I am not sure whether the solution obtained is a true solution to the original equations. (3). For me, the fight between FD and FE was long settled in 70's. It is really not a very interesting subject. And I don't think man and monkey are related at all. (4). In today's early moring news report, the reporters and the doctor were trying to show that man and woman think differently because their brain structure are different. I think, when a man thinks, the world is 3-D, while, when a woman thinks, the world is 2-D. That is why there is alway conflict between man and woman. The same applies to the FE, FD and FV. There will be questions asked over and over again, because they are all different things. Sure, they must have something in common.

hello Gary,

Order of accuracy of FV methods can be inproved by Flux Vector Splitting techinique, where one can use QUICK and MUSCL type schemes for higher order accuracy.

if you want to know more please refer to C. Hirsch, numerical computation of Internal and External flow,vol2, chapter21. one of our paper in this context is under revision. If you want, i will let you know once it get published.

regards,

Sridar D.

You can easily construct a higher order FEM method by using higher order test functions. Upwinding and other FVM tricks can also be contsructed in FEM by streamline-diffusion techniques etc. Please check papers by Tom Hughes and others for further info. The Hirch book isn't exactly the right reference for FEM! I think that FVM are more popular in fluid mechanics since most people in this field do not have a mathematical background. Besides, the major advantages with FEM is probably that you for some applications can have strict a'posteriori error control and that you can base adaptivity on this. However, there is no such thing as an a'posteriori error estimate for Navier-Stokes, so no error control for fluids problems!

As I see it the difference between FEM and FVM is the way you *derive* the method. Then you can always pick a specific scheme of one type and say that it is more accurate than another specific scheme of another type, but that doesn't say very much.