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I am trying to locate suitable computational algorithms for a optimization problem that requires repeated solution of transient 2D incompressible Euler equation on a 2D domain (say rectangular).

My question is whether an off-the-shelf PDE solver can solve incompressible Euler equations on 2D domains in a robust manner, or if there are any customized solvers for 2D incompressible Euler.

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    $\begingroup$ The solution to your problem doesn't require a computer code -- it is $\mathbf u_0=0$ since then the velocity field is and remains zero, and the points $\mathbf x_i$ do not move -- making your objective function zero. Is this what you had in mind? $\endgroup$ – Wolfgang Bangerth Oct 9 '17 at 22:14
  • $\begingroup$ @WolfgangBangerth Thanks for noticing the error. I edited my question so it makes more sense $\endgroup$ – mystupid_acct Oct 9 '17 at 23:39
  • $\begingroup$ I see. I don't know of any code that could do this, and I think it's also a really difficult problem to solve since there are likely many local solutions to the minimization problem -- think of rotating the imaging by 180 degrees in your disk domain: you can choose an initial condition that rotates clockwise or counterclockwise, and you can also rotate by 540, 900, ... degrees. $\endgroup$ – Wolfgang Bangerth Oct 10 '17 at 12:59
  • $\begingroup$ @WolfgangBangerth I am fine with local optimal solutions. My main question is about the Euler solver: are there some available Euler-equation solvers I can use that can be used to reliably generate particle trajectories needed for evaluating the terms in cost. $\endgroup$ – mystupid_acct Oct 10 '17 at 15:09
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    $\begingroup$ Why not formulate the adjoint to find the gradient of your cost function? Your design or forcing would be the initial condition. $\endgroup$ – Spencer Bryngelson Oct 12 '17 at 4:29
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Since you are solving inviscid equations, you will need some form of stabilization like SUPG or a DG scheme, to get a robust scheme. You wont need the pressure I presume. I would recommend solving the equations in vorticity-velocity form where the pressure is eliminated. See this paper for a DG scheme

Jian-Guo Liu and Chi-Wang Shu, A High-Order Discontinuous Galerkin Method for 2D Incompressible Flows, Journal of Computational Physics 160(2):577-596, August 1999, DOI: 10.1006/jcph.2000.6475

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