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Mimicking the Kuramoto-Sivashinsky Equation Using Cellular Automaton

time step
31
The KuramotoSivashinsky 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
u
4
x
+
4
u
4
y
+αu
u
x
+
2
u
2
x
+α
u
y
+
2
u
2
y
=0
,
where the diffusion term is
α
u
xx
+α
u
yy
, the dissipation term is
u
xxxx
+
u
yyyy
, and the advection term is
αu
u
x
+αu
u
y
;
u(x,y,t)
can represent any physical characteristic like velocity or a mixture fraction.
The initial setup assumes that there is a highly flammable center (the red area). An
L×L
square lattice is used with discretization in both
x
and
y
directions as
ΔL
. A von Neumann neighborhood is used. Energy contained in a lattice of size
r
is denoted by
E(r)
. At each time step, a unit amount of energy is added to a random lattice size as
E(
r
0
)E(
r
0
)+1
. If
E(
r
0
)E(
c
1
)
, then
E(
r
0
)0
; 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
E(
c
1
)
is the same for all sites and is time independent. The dissipation process is called burning.
Define the propagation energy threshold as
E(
c
2
)E(
c
1
)
. Neighbors of the burning site burn if their energy is greater than that of
E(
c
2
)
; that is, if
E
r
0
E(
c
0
)
, then
E
r
0
0
.
Reflecting boundary conditions are used.
Energy diffusion takes place at every time step as
E(r)E(r)+ΔE(r)
, where
ΔE(r)
is calculated as
D
Δt
Δ
2
L
r'
[E(r',t)-E(r,t)]+E(r,t)
, for all
r
,
where
E(r,t)
is the energy content of the site
r
at time
t
and the sum is over all the neighbors of
r
.
D
is the diffusion constant.
The advection term was not implemented as the authors are more interested in using the KuramotoSivashinsky equation in studying forest fires and simple combustions.
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