Introduction to the Modeling and Analysis of Complex Systems

250 CHAPTER 13. CONTINUOUS FIELD MODELS I: MODELINGspatial discretization is as follows:∂f∂x ≈ 2∆f2∆x=f(x + ∆x, t) − f(x − ∆x, t)2∆x(13.33)This version treats the left and the right symmetrically.Similarly, we can derive the discretized version of a second-order spatial derivative, asfollows:∂f∂ 2 f∂x ≈ ∂x2∣ − ∂f∣x+∆x∂x∣x−∆x2∆xf(x + ∆x + ∆x, t)−f(x + ∆x − ∆x, t)f(x − ∆x + ∆x, t)−f(x − ∆x − ∆x, t)≈2∆x−2∆x2∆xf(x + 2∆x, t) + f(x − 2∆x, t) − 2f(x, t)=(2∆x) 2f(x + ∆x, t) + f(x − ∆x, t) − 2f(x, t)=∆x 2 (by replacing 2∆x → ∆x) (13.34)Moreover, we can use this result to discretize a Laplacian as follows:∇ 2 f = ∂2 f∂x 2 1+ ∂2 f∂x 2 2+ . . . + ∂2 f∂x 2 n≈ f(x 1 + ∆x 1 , x 2 , . . . , x n , t) + f(x 1 − ∆x 1 , x 2 , . . . , x n , t) − 2f(x 1 , x 2 , . . . , x n , t)∆x 2 1+ f(x 1, x 2 + ∆x 2 , . . . , x n , t) + f(x 1 , x 2 − ∆x 2 , . . . , x n , t) − 2f(x 1 , x 2 , . . . , x n , t)∆x 2 2+ . . .+ f(x 1, x 2 , . . . , x n + ∆x n , t) + f(x 1 , x 2 , . . . , x n − ∆x n , t) − 2f(x 1 , x 2 , . . . , x n , t)∆x 2 n(= f(x 1 + ∆x, x 2 , . . . , x n , t) + f(x 1 − ∆x, x 2 , . . . , x n , t)+ f(x 1 , x 2 + ∆x, . . . , x n , t) + f(x 1 , x 2 − ∆x, . . . , x n , t)+ . . .+ f(x 1 , x 2 , . . . , x n + ∆x, t) + f(x 1 , x 2 , . . . , x n − ∆x, t))/− 2nf(x 1 , x 2 , . . . , x n , t) ∆x 2 (13.35)(by replacing ∆x i → ∆x for all i)