Difference between Stencil -structures and Cellular Automata Category-theoretically? Definitions
  
  Stencil =
  "For a given point, a stencil is a pre-determined set of nearest
  neighbors (possibly including itself)."
  (source)
  
  Wikipedia's definition (source) =
  


It looks like CA (cellular automata) is a broader (and perhaps more sexy fuzzy populist) concept while the stencil (S) is more rigorous. Suppose a torus -topology in a grid and a simple structure where each state depends only on the next state (some transition rule, 4 neighbors but only next state matters). This I think is a Stencil, Cellular automaton, a grid, a mesh, easily described with Markov Chain -- and a structure. Now I am confused! So many different words for the same thing? Please, add references to quantify this concept, many authors seem to use different words for the same thing -- not sure what it is, some structure with function -definitions -- perhaps there exists some easy category-theoretic description for them?

If this is too elementary question, I would be happy about some reference covering definitions rigorously and doing some comparisons.

Perhaps related


Automatic Stencil Code Generation Ph.D. Thesis Proposal here. The author behind the last paper here, researching something about stencil things and a lot of perhaps related publications here, related to the last author.
Fey, Dietmar et al. (2010) Grid-Computing: Eine Basistechnologie für Computational Science. Page 439. (Source)
Complex Systems from the Perspective of Category
Theory: I. Functioning of the Adjunction Concept here
Judging from the source you provided, there appear to be two key differences:
(1) stencils have numeric values in each cell and update the cells with functions defined numerically (typically using continuous functions) while cellular automata typically have only a finite set of values per cell and use functions that in general do not arise from numerical methods, and (2) stencils are updated by a sweep across the grid while in cellular automata generally all cells are updated simultaneously.