Hello everybody on forum

Can anyone clarify difference between Finite Difference method and Finite Volume method and which method gives efficient results.

sameer mohrir

Finite volume method lets conservation of a variable in time and space by integrales but finite difference method can't assure this requirement.

for complete information try to look this book

An Introduction to Computational Fluid Dynamics: The Finite Volume Method

By: Versteeg, H.K. and Malalasekera, W. Addison-Wesley, 1996

(1). If you write the governing equations in differential form and solve the derived algebraic equations as a result of discretization using Taylor series expansion, then it is finite-difference approach. This is the right approach. (2). If you write the governing equations in integral form for a small control volume and solve the derived algebraic equations as a result of approximation to the integration, then it is finite-volume approach. (3). So, in general, the finite-volume approach does not satisfy the governing equations in differential form. But it gives you the global conservation of mass, momentum,etc...(4). If the geometry is very simple, and the mesh is simple Cartesian, it can be shown that both approaches give the same algebraic equations, thus the same answers. For general geometry, this is not the case. (5). Since we are normally looking for the mesh independent solution, the mesh size has to be systematically reduced to reach this goal, both methods should give the same result in the end. (6). If the algebraic equations are the same, then the efficiency (speed of getting the solution) is the same. (7). To some extent, the finite-difference approach is a field method, while the finite-volume approach is a particle method. (8). The finite-element approach is a completely different formulation, which is valid at the finite size element. So, in theory, there is no mesh independent solution requirement for the finite-element approach. Any solution is a valid solution, although they are not the same.

Hello John

Thanks for clarification

sameer

I also have a question about the differences of FV and FD.In Finite Volume,the dependent variables are cell-average value,but the experiment results usually are the point values,which are difficult to integrate it to cell-average value , so that we can't simply compare the cell-average value with the point value. However,in many papers,the results of finite-volume usually are compared with the experiment.Is this correct?If not so,How can we get the cell-average value form the experiment result;If so,why?

(1). It is likely that the mesh they used have cell size smaller than the probe volume size. So, in this case, there is no resolution inside the probe volume. (2). But this is just my theory. For specific papers, you need to read the mesh size they used in the analysis, and also the measurement technique in order to make comparison. (3). If the cell size is much larger than the probe size, then, all we can say is that he is presenting the test data as a reference. (4). Also, if the cell center value represent the average value of the cell, then the distribution of the computed variable will be discontinuous across the cell boundary. (step-wise distribution). (5). I must say that you don't have to make everything uniform inside a cell, although it is one of the possibility. (6). As I mentioned before, both finite-difference and finite-volume approach must satisfy the mesh independent requirement, by mesh refinement, therefore, the right approach is to compare the mesh independent solution with the test data (assmuning that the probe volume is very small. (this can easily become a problem for the test data in a boundary where the thickness of the boundary layer is relatively thin. (7). So, getting a solution from a finite-volume code is one thing, making the comparison with the test data is another thing. Both must be carefully executed. This also reveals the limitations on the scale model testing. And sometimes, one is forced to use a scaled up model in order to obtain useful data due to the probe volume limitation. (8). For validation test, in most cases, it is desirable to plan ahead so that the computed value is located at the same place where the measurement is carried out. In this way, it will eliminate the need to do interpolation. (it is more difficult to follow this principle when using unstructured mesh, because the cell location is hard to control)

The difference between a cell-averaged value and a point value at the centroid of the cell is a second order quantity. If the method is 2nd order accurate, it doesn't matter which value you're computing. For higher order method, it may be a problem.

So, What's the best method between FVM and FDM, or FEM ?

If we run a CFD code on the grid with out any 'body', then we should only get free stream flow in the entire domain. In some cases this does not happen for FDM codes. I guess by using approximations to the derivatives in FDM we tend to intoduce 'Free stream errors'. Finite volume codes do not have this problem.

Finite element methods (FEM) typially use a weighting function across cells to arrive at the approximation for a cell. FVM is a special case of FEM where this weighting function is a constant. As mentioned on Cartesian gris all methods FDM, FVM and FEM give identical results.

An FDM scheme in conservative form on a structured grid can be shown to be a FVM and vice versa. Obviously an FV scheme can be written on an unstructured grid. It is not accurate to say that FVM is a special case of FEM where the basis function is a constant. The latter is first order accurate and clearly FVM can be accurate to any desired order. Of course conservation is guaranteed if the sum:n(dot)ds evaluated over the entire domain is zero (n is the oriented surface normal, ds the surface area of the face of each elemental volume).

Since one can have a stretched cartesian grid, FDM,FVM and FEM will be equivalent on a uniform cartesian grid.

Ravichandran

I think it is true in speculating that FVM is not strictly a sub-class of FEM with a piecewise constant basis.

However, it should be noted that while constructing the solution on a FEM mesh the nodal values can be constructed to any order of accuracy and this is in some way related to MUSCL/TVD implementations with a piecewise constant construction for the element. This would be a discontinuous Galerkin formaulation.

However, I think that FEM studied from a Functional Analysis perspective is far more mathematically rigorous in its perspective with strong theorems built around Hilbert Spaces which can ascertain a scheme's robustness. It should be noted that FVM is basically built with integrals which in L2 space deals with functionals and this can be strongly co related with the rich F.A. background of FEM.

Pointers in this direction are due to books by Strang and Fix, and M. Gunzberger.

Ravi,

Thanx for your response. What I meant was that FV schemes of higher orders cannot be obtained formally from an FE formulation with zeroth order basis functions. In fact the choice of the basis function space will depend on the number of nodes as well as the degrees of freedom associated with each node. Obviously parallels do exist among higher order schemes of both family.

Yes, FE formulations lend themselves easily to mathematical analysis since they can be embedded in appropriate function spaces and the tools of functional analysis play an effective and powerful role in obtaining estimates. Also thanx for the references.

Ravichandran