Introduction to the Modeling and Analysis of Complex Systems

13.5. SIMULATION OF CONTINUOUS FIELD MODELS 251The numerator of the formula above has an intuitive interpretation: “Just add the valuesof f in all of your nearest neighbors (e.g., top, bottom, left, and right for a 2-D space) andthen subtract your own value f(x, t) as many times as the number of those neighbors.”Or, equivalently: “Measure the difference between your neighbor and yourself, and thensum up all those differences.” It is interesting to see that a higher-order operator likeLaplacians can be approximated by such a simple set of arithmetic operations once thespace is discretized.You can also derive higher-order spatial derivatives in a similar manner, but I don’t thinkwe need to cover those cases here. Anyway, now we have a basic set of mathematicaltools (Eqs. (13.31), (13.33), (13.34), (13.35)) to discretize PDE-based models so we cansimulate their behavior as CA models.Let’s work on an example. Consider discretizing a simple transport equation withoutsource or sink in a 2-D space, where the velocity of particles is given by a constant vector:∂c= −∇ · (cw) (13.36)∂t( )wxw =(13.37)w yWe can discretize this equation in both time and space, as follows:c(x, y, t + ∆t) ≈ c(x, y, t) − ∇ · (c(x, y, t)w) ∆t (13.38)⎛ ⎞∂ ( ( ))⎜= c(x, y, t) − ⎝∂x ⎟wx∂ ⎠Tc(x, y, t) ∆t (13.39)w y∂y()∂c= c(x, y, t) − w x∂x + w ∂cy ∆t (13.40)∂y(c(x + ∆h, y, t) − c(x − ∆h, y, t)≈ c(x, y, t) − w x2∆h)c(x, y + ∆h, t) − c(x, y − ∆h, t)+w y ∆t (13.41)2∆h= c(x, y, t) − ( w x c(x + ∆h, y, t) − w x c(x − ∆h, y, t)+ w y c(x, y + ∆h, t) − w y c(x, y − ∆h, t) ) ∆t2∆h(13.42)This is now fully discretized for both time and space, and thus it can be used as astate-transition function of CA. We can implement this CA model in Python, using Code11.5 as a template. Here is an example of implementation: