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.