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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 $(a\cos(r), b\sin(r)$). 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 $u(s,0)=\cos^2(2\pi s/L)$. The solution should be $u(s,t)=\cos^2(2\pi(s-t)/L)$. 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.

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    $\begingroup$ Why is the curve shape important at all? Isn't it the same as solving on a straight 1D periodic domain? $\endgroup$ – Maxim Umansky Sep 9 at 22:48
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    $\begingroup$ I agree with @MaximUmansky. I think that it's the same as a 1d periodic domain for which you can write down the analytic solution (which I believe you've actually already done, though with a typo -- the time is missing on the lhs). $\endgroup$ – Wolfgang Bangerth Sep 10 at 2:45
  • $\begingroup$ Thank you, I see now that the solution is just the same as substituting say $s=x$ for a normal pde. I guess my main problem is the overall formatting - I originally had the general advection equation on a manifold, which from my understanding for a curve simplifies to the derivative (or 2nd derivative for the laplace beltrami operator) with respect to arc length. I'm wondering generally how I can relate the solution to the curve x(r), y(r), so that I can visualize the solution on the curve rather than the solution just being dependent on arc length. I'm also $\endgroup$ – lrs417 Sep 10 at 12:54
  • $\begingroup$ Assuming you have the solution as a function of the arc length $s$, and you have the parametric representation of the curve $x(s), y(s)$, you have all what's needed to plot the solution as a function of the spatial coordinates. $\endgroup$ – Maxim Umansky Sep 10 at 14:54
  • $\begingroup$ Would you be able to give an more insight as to how I would do this given x(s), y(s)? I have the solution u(s,t), and I want to plot the solution in cartesian coordinates along with the curve. $\endgroup$ – lrs417 Sep 10 at 17:33

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