What I am writing here is mostly just a digest of other users' comments above. As it stands, the short

Have cellular automata been applied to solving PDEs on closed curves? If yes, what are some recommended reads on the topic that develop the theory?

it not a good question for MathOverflow. It is currently too broad, and it is unclear whether it admits an answer, and it sounds a bit like a fishing expedition since, as vzn wrote

CAs are digital, how can they be applied to continuous curves?

On the other hand, if you edit the question a bit it may become a good question.

**Be more precise**: one of the easy ways to address vzn's comment is to restrict your attention to the case of *numerically solving PDEs*. As numerical methods are by definition digital, such an objection cannot arise. But please note that if you are asking a question about the frontiers of numerical methods, the website Computational Science may have more experts with up-to-date knowledge.

In addition, what exactly to you mean by closed curves? What sorts of PDEs are you thinking about? What benefits are you envisioning coming from CA?

For example, if you mean by closed curves the smooth image of $\mathbb{S}^1$ in some manifold, then why are you even thinking about PDEs? Shouldn't the relevant question be one of *ordinary* differential equations? Do you actually mean closed algebraic curves? Then can your problem be described in the context of differential algebra? Is the word "curves" a typo? Are you thinking about some sort of fractal? or some sort of higher dimensional object?

Next, what kinds of PDEs are you thinking about? If the PDE admits any sort of maximum principle, then on a closed manifold it cannot admit anything other than the trivial solution.

**What do you know?** Why are you led to ask this question? Are you faced with a particular PDE that cannot be easily resolved using conventional methods? Does the PDE itself has a good physical/computational interpretation that makes the CA implementation sensible? Do you know of any cases where CA has been applied to study any ultimately *continuous* phenomena? Are there examples where CAs have been shown to be useful for solving a PDE, not necessarily on a closed curve? Have you heard of this idea from someone in particular and want to ask for clarification?

You wrote that the question is inspired by a **research problem** you are struggling with. Please share as much about the actual problem you are working on as possible. (I understand there may be some desire to keep some details under wraps: but do understand that as mathematicians, we are *trained* for abstraction. Perhaps your CA idea is complete hogwash, but it is possible if you undo some of your own abstraction and show us something a bit more raw, we can give you something else, which may not be CA, but which may still be useful.)

(As an aside, this reminds me of the (possibly apocryphal) story of the engineer who walked into the office of a mathematician and asked to know the general form of the Schwarz-Christoffel mapping onto an arbitrary polygon. The mathematician dutifully wrote down the answer and asked about the reason for the question. The engineer's reply was allegedly: "Well, we really want to know the case for the mapping onto a round disc, but we figured that question is too hard, so we'll just ask you the easy one and approximate by taking $n$ to infinity.")