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.