Solving space-dependent equations using dealII?

[Maxi Miller]

Is it possible to solve not only time-, but also space-dependent equations in dealII? As example the nonlinear schrödinger equation:
\partial_z U = i(\partial_x^2+\partial_y^2) U
with U=U(x, y, z) and U_0=U(x, y, 0)? As far as I know until now, all the equations are based on time, but stationary in space. Or are there similar examples which I did not see yet in the tutorial?
Thanks!

[Daniel Arndt]

Maxi,

Is it possible to solve not only time-, but also space-dependent equations in dealII? As example the nonlinear schrödinger equation:
\partial_z U = i(\partial_x^2+\partial_y^2) U
with U=U(x, y, z) and U_0=U(x, y, 0)? As far as I know until now, all the equations are based on time, but stationary in space.

What exactly do you mean by "stationary in space"? The interpretation of derivatives is totally up to you.

What is the difference between the equation above and the instationary equation

  \partial_t U = i(\partial_x^2+\partial_y^2) U

U=U(x,y,t) and U=0(x,y,0) you are considering in https://groups.google.com/forum/#!topic/dealii/1RQE2TpZT9I ?

Best,

Daniel

[Maxi Miller]

I think my question was badly formulated. When solving the equation above, I get 2d-slices for every step, which are then assembled to a 3d-model in z-direction. Is something similar possible with dealII, or do I have to use external programs for that?
And yes, the same can be done with time instead of z as position.

[Wolfgang Bangerth]

On 08/22/2017 09:30 AM, 'Maxi Miller' via deal.II User Group wrote:
> I think my question was badly formulated. When solving the equation
> above, I get 2d-slices for every step, which are then assembled to a
> 3d-model in z-direction. Is something similar possible with dealII, or
> do I have to use external programs for that?

You can do either. You can either treat 'z' as a time-like variable and
do time-stepping in this direction (solving a sequence of 2d problems in
the process) or use a single problem in the same vein as people do when
they use space-time finite elements. If you search for "space-time
finite element methods", you will find a lot of literature -- in
essence, you end up with a problem that is diffusive in one direction
but an advection problem in the time/z direction; as always for
advection problems, you will have to choose an appropriate formulation
to make things stable.

Both methods are of course implementable in deal.II.

Best
W.


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

[Daniel Arndt]

Maxi,

adding to Wolfgang's answer: if you only want to visualize all time steps (or the z-dependency)
at once, you can use DataOutStack[1] for this.

Best,
Daniel

[1] https://www.dealii.org/8.5.0/doxygen/deal.II/classDataOutStack.html

[Maxi Miller]

That class looks very useful, thank you!