Time derivative on right hand side for equation

[Maxi Miller]

I have an equation system where the right hand side contains a time derivative, looking like

Without that time derivative I can solve it, but I do not know how to include it here. Are there possible guidelines for that? One idea I got is from here, but this can not applied to the code in my case (used example 52 as starting point), as I assume.

[Wolfgang Bangerth]

On 07/18/2017 07:21 AM, 'Maxi Miller' via deal.II User Group wrote:
> I have an equation system where the right hand side contains a time
> derivative, looking like
>

> <https://lh3.googleusercontent.com/-zRtosh4pbeY/WW4KxK2_2qI/AAAAAAAACPQ/Ln8NbS9GxMQ5JUYTQIbuj7g19BvGh9mnwCLcBGAs/s1600/time_derivative.png>

>
>
> Without that time derivative I can solve it, but I do not know how to include
> it here. Are there possible guidelines for that? One idea I got is from here

> <https://scicomp.stackexchange.com/questions/27363/approach-for-coupled-equations-with-included-time-derivative>,

> but this can not applied to the code in my case (used example 52 as starting
> point), as I assume.

Why can't this approach be used here as well? You can still use the first
equation to substitute the time derivative in the second equation.

Best
W.

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

[Maxi Miller]

So, you suggest to calculate first the first equation, and then use the result for the second equation?

[Wolfgang Bangerth]

On 07/18/2017 09:00 AM, Maxi Miller wrote:
> So, you suggest to calculate first the first equation, and then use the result
> for the second equation?

No -- I suggest substituting
d/dt f(A)
by
f'(A) d/dt A
and then using the first equation to substitute d/dt A in that equation. Just
get rid of the time derivative altogether.

[Maxi Miller]

That would result in (including my starting equations):

But how do I then get the value for A after each iteration?
Thank you!

[Wolfgang Bangerth]

On 07/18/2017 12:52 PM, Maxi Miller wrote:
> That would result in (including my starting equations):
>

> <https://lh3.googleusercontent.com/-j1YR1Gju6b8/WW5YkX74UMI/AAAAAAAACPw/_oAfa2Oc7TkfDCfDU-ne5yWOoPQY99PzACLcBGAs/s1600/time_derivative_II.png>

>
> But how do I then get the value for A after each iteration?

That is a question for the time stepping scheme you use. If you use something
like step-21, you first solve the first equation for A, then put the result
into the second equation. If you use a semi-implicit scheme for the coupled
problem, you take the previous time step's value of A for the right hand side.
If you take a fully coupled scheme, you'd get a nonlinear system that you have
to solve for both A and B at the same time.

This is really no different than the case where A,B were only dependent on the
time. Think about how you would write an ODE solver in that case.