[anon_q]

Hello,
while reading a CFD book, I came across this paragraph:

... For the coarse mesh, the longest possible wavelengths of error components (i.e. those of the order of the domain size) are just within the short-wavelength range
of the mesh and, hence, all error components reduce rapidly. On the finer meshes, however, the longest error wavelengths are progressively further outside the short-wavelength range for which decay is rapid.

• What does exactly mean error wavelength?
• Could you please give me an example?

Thank you

[FMDenaro]

Originally Posted by Evren Linda

Hello,
while reading a CFD book, I came across this paragraph:

• What does exactly mean error wavelength?
• Could you please give me an example?

Thank you

consider the error like a function built by a Fourier series, a coarse mesh is able to do a "filtering" effect on the components of the errors. When the mesh is sufficiently coarse also the components at low wavenumbers are smoothed. Concersely, for a fine mesh you have a very slow decay.
This is a concept at the basis of the multigrid methods

[LuckyTran]

Consider a cell and its immediate adjacent neighbors. You do the FVM method on this cell and try to solve your equations but you end up with a small error (the residual). This is the short wavelength error.

Now you have a cell in the inside the domain (say the center). You end up with a solution for this interior cell, whatever that solution is. Although it may (or may not) satisfy the local flux balances, it may be inconsistent with the boundary conditions imposed on the domain surfaces. This is the long wavelength error.

There are intermediate wavelengths... say between a cell and not its immediate neighbors but a few houses down.

As already mentioned, the grid acts like filter and a smoother. Because of the way the FVM method tends to localize the solution process (the linear system that you solve usually consists of a cell and its nearest neighbors), the longest wavelength errors don't get removed very quickly compared to the smallest wavelength errors. As the grid becomes finer it becomes worse.

The practical outcome of this phenomenon is, without multi-grid algorithms or convergence accelerators, you need more and more iterations to converge as you make your mesh finer and finer (in addition to the calculation already being more computationally costly on a finer mesh).