# Robust/Tested Solver for incompressible 2D Euler (Fluid dynamics) Equation

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.

• 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? Commented Oct 9, 2017 at 22:14
• @WolfgangBangerth Thanks for noticing the error. I edited my question so it makes more sense Commented Oct 9, 2017 at 23:39
• 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. Commented Oct 10, 2017 at 12:59
• @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. Commented Oct 10, 2017 at 15:09
• Why not formulate the adjoint to find the gradient of your cost function? Your design or forcing would be the initial condition.
– user20857
Commented Oct 12, 2017 at 4:29