# Mimicking the Kuramoto-Sivashinsky Equation Using Cellular Automaton

Mimicking the Kuramoto-Sivashinsky Equation Using Cellular Automaton

The Kuramoto–Sivashinsky system arises in the description of the stability of flame fronts, reaction-diffusion systems, and many other physical settings. It is a simple nonlinear PDE that exhibits chaotic behavior in time and space. The equation was introduced as a model of instabilities on interfaces and flame fronts by Sivashinsky and as a model of phase turbulence in chemical oscillations by Kuramoto. The equation in 2D is given as

∂u

∂t

4

∂

∂

4

x

4

∂

∂

4

y

∂u

∂x

2

∂

∂

2

x

∂u

∂y

2

∂

∂

2

y

where the diffusion term is , the dissipation term is +, and the advection term is ; can represent any physical characteristic like velocity or a mixture fraction.

α+α

u

xx

u

yy

u

xxxx

u

yyyy

αu+αu

u

x

u

y

u(x,y,t)

The initial setup assumes that there is a highly flammable center (the red area). An square lattice is used with discretization in both and directions as . A von Neumann neighborhood is used. Energy contained in a lattice of size is denoted by . At each time step, a unit amount of energy is added to a random lattice size as . If , then ; that is, if the energy at a lattice point is greater than some threshold energy, the energy from that lattice point gets dissipated. The threshold energy is the same for all sites and is time independent. The dissipation process is called burning.

L×L

x

y

ΔL

r

E(r)

E()→E()+1

r

0

r

0

E()≥E()

r

0

c

1

E()→0

r

0

E()

c

1

Define the propagation energy threshold as . Neighbors of the burning site burn if their energy is greater than that of ; that is, if , then .

E()≤E()

c

2

c

1

E()

c

2

E≥E()

r

0

c

0

E→0

r

0

Reflecting boundary conditions are used.

Energy diffusion takes place at every time step as , where is calculated as , for all ,

E(r)→E(r)+ΔE(r)

ΔE(r)

D[E(r',t)-E(r,t)]+E(r,t)

Δt

Δ

2

L

∑

r'

r

where is the energy content of the site at time and the sum is over all the neighbors of . is the diffusion constant.

E(r,t)

r

t

r

D

The advection term was not implemented as the authors are more interested in using the Kuramoto–Sivashinsky equation in studying forest fires and simple combustions.