I read the paper "Stable, circulation-preseving simplicial fuids" by Elcott, et al: http://www.cs.jhu.edu/~misha/Fall09/Elcott07.pdf. It gives a structure preseving discretization of fluids. I have a question about this paper in its "advect vorticity" step (page 7, section 2.3). It shows (in my words) "we can backtrack all the dual vertices, then compute the corresponding circulation". But there are some dual faces (namely, those on the boundary), whose boundary does not comprise dual edges only. They also comprise some line segments that completely lie on the boundary face (easy to see if you draw a graph). In this case, we are not able to find the vorticity on these faces by backtracking dual vertices only. We may need to backtrack more points (for example, the centers of boundary faces and edges) so that we can compute the circulation then.
The above is my understanding. I am not sure, but it seems the paper didn't talk about this. I am just wondering if anyone who read this paper can give me some help!