Courant Number for Finite Volume Method
 

Does anybody know if there is a difference between the Courant Number equation for the finite difference method and the finite volume method?

A book title with the equation would be enough.

thanks

https://books.google.it/books/about/...er&redir_esc=y

https://books.google.it/books/about/...ir_esc=y&hl=it

When do i need the folowing equation?



https://openfoamwiki.net/images/math...d5a50a961b.png

And what does it mean?

is there Delta V the cell volume and Delta S a area from the cell surface?

found it on:

https://openfoamwiki.net/index.php/Co

Well that means OpenFOAM is calculating the Courant Number as a product
of the ratio of the sum of the flux trough the cell faces per cell volume and the timestep width.




But what means the factor 0.5.

Most probably it is trying to obtain minimum edge length from the available control volume surface areas and volumes. The absolute control volume area summation close to walls is mainly dominated by lower and upper control volume surface area values. Therefore, the area summation should be divided by 2. Then minimum edge length will be approximately Volume/(0.5 Area sum).

The 1/2 is OpenFOAM specific and has to do with the way OpenFOAM calculates "the summation over all faces." The way it is done in OpenFOAM, it actually double counts because there is a face/flux belonging to the current cell and a face/flux belonging to the neighboring cell and both get summed in the summation.

OpenFOAM calculates fluxes using a loop over faces and add/subtract flux vector value to right and left cells, respectively. This would be i and j vertices for a vertex based method FVM. When the loop is finished, all control volume faces are done. This does not change the face flux value. But it significantly reduce the computational cost (almost by half).

CFL number may be considered as the time required to move out/replace the conserved variable trough control volume surfaces with either positive or negative fluxes (incoming or outgoing) for FVM approximation (CFL=1). The absolute summation gives two times of this flux (for steady state, this flux summation without absolute value is zero). But one can sum negative and positive flux, and then take the maxımum absolute value. Then no need for 1/2.