When I read about the Church-Turing thesis it seems to be a common claim that "physical reality is Turing-computable." What is the basis for this claim? Are there any theoretical results along these lines?
For context, I am a researcher who works on physical simulations, so of course I'm aware that many partial differential equations (PDEs) that would arise in nature (e.g. the heat equation, wave equation, etc) can be approximated by numerical methods like finite elements, and that for many PDEs a solution can be estimated to arbitrary accuracy given enough computation (by decreasing the space and time step sizes).
However, I also know that proving convergence of finite element methods is notoriously difficult for PDEs of any appreciable complexity, even "easy" PDEs like the mean curvature flow that describes the shape of a soap film. I also know that many "Zeno-type" situations arise in practice in physical systems, such as Euler's disk or inelastic collapse. Is there reason to believe that the solutions to all PDEs, or at least all PDEs that would arise in nature, are Turing-computable?