Morphology of Experimental and Simulated Turing Patterns

The advantage of the FTCS method is that it is easy to implement. The method isconditionally stable for values of ∆t and ∆ given by Eq. (3.31), which is obtainedby a von Neumann stability analysis in section 3.3.However the FTCS schemes generally converge more slowly than implicit methods.Therefore a semi-implicit Crank-Nicolson scheme is discussed in the followingsection.However due to the system-size dependence of the semi-implicit scheme the FTCSmethod has been used for the generation of most patterns in this thesis.3.2 Semi-implicit Crank-Nicolson schemeFor a two-dimensional diffusion equation a FTCS scheme might not be the rightchoice, as shown in Ref. [50], as a large number of small time-steps has to be calculatedto evolve up to a timescale of physical interest. The implicit Crank-Nicolsonmethod gives an unconditionally stable solver for this special case. A scheme iscalled explicit if the solution for the time-step n + 1 is given as an explicit functionof the solution at time n, and implicit if it requires the solution of a system of linearequations for each time-step. A semi-implicit scheme for a general reaction-diffusionsystem, as given by Eq. (3.1) and Eq. (3.2) can be obtained similarly by discretizingthe spatial derivative as an average of forward and backward Euler in time:∇ 2 u = 1 2( unj+1,l+ u n j−1,l + un j,l+1 + un j,l−1 − 4un j,l(∆) 2 +u n+1j+1,l + un+1 j−1,l + un+1 j,l+1 + un+1 j,l−1 − )4un+1 j,l(∆) 2 + O(∆ 4 ).(3.9)For the time derivative the same first order discretization is used as for the FTCSscheme:∂u∂t ∣ = un+1 j,l− u n j,l+ O(∆t 2 ). (3.10)j,l∆tUsing Eq. (3.9) and Eq. (3.10) to discretize the reaction-diffusion system describedby Eq. (3.1) and Eq. (3.2) and separating the {u,v} n and {u,v} n+1 terms yields(1 + 2γ u ) u n+1j,l− γ ()uu n+12j+1,l + un+1 j−1,l + un+1 j,l+1 + un+1 j,l−1=(1 − 2γ u ) u n j,l + γ u(un2 j+1,l + u n j−1,l + un j,l+1 + ) ( (3.11)un j,l−1 + ∆t f unj,l ,vj,l) n ,and(1 + 2γ v ) v n+1j,l(1 − 2γ v ) v n j,l + γ v2− γ ()vv n+12j+1,l + vn+1 j−1,l + vn+1 j,l+1 + vn+1 j,l−1=(vnj+1,l + vj−1,l n + vn j,l+1 + ) ( (3.12)vn j,l−1 + ∆t g unj,l ,vj,l) n ,29