numerical solution to pde on an ellipse

Looking for advice on discretization (preferably finite difference) schemes for pdes on curves in general, but in this case it is an ellipse (so given by $(acos(r), bsin(r)$). The problem is the advection equation

$$frac{ partial u_s}{partial t}+ frac{ partial u_s}{partial s}=0,$$
with $s$ corresponding to arc length and the initial condition $u(s,0)=cos^2(2pi s/L)$. The solution should be $u(s,t)=cos^2(2pi(s-t)/L)$. I don’t want to do this using the closest point method, I want to keep the curve parameterized and just discretize on the curve but I’m not sure how to proceed.