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I have a 2D surface in 3D that I want to advect under a velocity field. More precisely, I have a surface $S$ and a velocity field $v$ and I want to advect $S$ under $v$ using the flow map of $v$, i.e. I want to compute $F^t(S_0)$, where $F^t$ is the flow of $v$. The velocity field is chaotic, so a direct computation leaves a bunch of scattered points. $S_0$ is very smooth.

I have been told that there are the so-called level set methods to do what I'd like. The idea sounds very nice, but this is not my background and it would be great if there are any codes available that do this.

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Level set methods are really only one particular case of advection PDEs which you can solve with any of the modern PDE toolboxes such as deal.II (disclaimer: that's my own library), fenics, libmesh, ... What makes it a level set method is that you the evaluate the solution function to find that surface where, for example, the solution equals zero. I'd try to look into one of these PDE toolboxes and see what you can use to solve your problem. They certainly all can solve advection equations.

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