Hi, if solving an incompressible flow using ACM method, for time dependent cases, we need to use dual time technique. Can some one explain me for example if I want to solve a simple question du/dt=d2u/dx2. I add a fictional time step as : du/dt1+du/dt=d2u/dx2.
What is the relation between fictious time step t1 and real time step t.
how can I implement this in a code?
Thanks
You may think of a numerical and physical time. The numerical time is for an inner loop from physical time level n to n+1 to reach a numerical steady state for n+1. The marching in numerical time is a kind of iteration to reach a "converged" solution for physical time level n+1. Since the marching in numerical time is to a steady state any time-inaccurate "convergence" acceleration methods can be used. See for instance the VKI report from 1986 on my RG page.
with no more explanation:
R(u^n+1)=0 could be solved by
du*/dt_1+R(u*)=0 //pseudo time term
.i.e
(u**-u*)/dt1+R(u*)=0
in unsteady
R(u)=du/dt2-d2u/dx2
therefore for example using first order in time we would have
(u**-u*)/dt1+(u**-un)/dt2-d2u*/dx2=0
u**=(u*/dt1+un/dt2+d2u*/dx2)/(1/dt1+1/dt2)
u* is the data in one iteration before and un is the data of one time step before.
when u** converged, it will be u^n+1
In my answer I referred to the VKI report from 1986 on my RG page. Unfortunately the file uploaded so far was not complete. Now it is, and you will find the basics of the method well described. Check whether the file is from April 19, 2016.
Technical Report A fully implicit scheme for unsteady flow calculations solve...
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