Please explain the projection method (tutorial step-35)

[Alexey Ozeritskiy]

Hello,

In step-35 we find u^{k+1} (diffusion step)

Then we find \phi^{k+1} (projection step) and correct p^{k+1} pressure. 

But why we don't correct u^{k+1} velocity? I think u^{k+1} must be solenoidal but it is not.

[Wolfgang Bangerth]

Alexey,
I'm not enough of an expert on projection methods, but you might want to try
the review paper by Guermond from a few years ago to get some of the
background of these methods. I don't know whether u^{k+1} must be solenoidal,
but it is quite possible that it doesn't have to be -- only that the
divergence is small (some power of h).

Best
W.


--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Alexey Ozeritskiy]

понедельник, 1 июня 2020 г., 6:14:49 UTC+3 пользователь Wolfgang Bangerth написал:

On 5/29/20 9:42 AM, Alexey Ozeritskiy wrote:
>
> In step-35 we find u^{k+1} (diffusion step)
> Then we find \phi^{k+1} (projection step) and correct p^{k+1} pressure.
> But why we don't correct u^{k+1} velocity? I think u^{k+1} must be solenoidal
> but it is not.

Alexey,
I'm not enough of an expert on projection methods, but you might want to try
the review paper by Guermond from a few years ago to get some of the
background of these methods. I don't know whether u^{k+1} must be solenoidal,
but it is quite possible that it doesn't have to be -- only that the
divergence is small (some power of h).

Thanks, the answer is in the papers (Guermond, 1995, eq 5.11, Guermond, 1998, eq 3.5). Actually we find and use non-solenoidal (intermediate) u^k on the diffusion step. 

"The _projected_ velocity can be completely _eliminated_ from the algorithm and the _intermediate_ velocity is the only one of interest." (Guermond, 1998)

As I understand, we don't need the projected (solenoidal) velocity for the calculation.