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Article overview
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Front propagation in reaction-diffusion systems with anomalous diffusion | | D. del-Castillo-Negrete
; | | Date: |
10 Sep 2014 | | Abstract: | A numerical study of the role of anomalous diffusion in front propagation in
reaction-diffusion systems is presented. Three models of anomalous diffusion
are considered: fractional diffusion, tempered fractional diffusion, and a
model that combines fractional diffusion and regular diffusion. The reaction
kinetics corresponds to a Fisher-Kolmogorov nonlinearity. The numerical method
is based on a finite-difference operator splitting algorithm with an explicit
Euler step for the time advance of the reaction kinetics, and a Crank-Nicholson
semi-implicit time step for the transport operator. The anomalous diffusion
operators are discretized using an upwind, flux-conserving, Grunwald-Letnikov
finite-difference scheme applied to the regularized fractional derivatives.
With fractional diffusion of order $alpha$, fronts exhibit exponential
acceleration, $a_L(t) sim e^{gamma t/alpha}$, and develop algebraic decaying
tails, $phi sim 1/x^{alpha}$. In the case of tempered fractional diffusion,
this phenomenology prevails in the intermediate asymptotic regime
$left(chi t
ight)^{1/alpha} ll x ll 1/lambda$, where $1/lambda$ is
the scale of the tempering. Outside this regime, i.e. for $x > 1/lambda$, the
tail exhibits the tempered decay $phi sim e^{-lambda x}/x^{alpha+1}$, and
the front velocity approaches the terminal speed $v_*=
left(gamma-lambda^alpha chi
ight)/ lambda$. Of particular interest is
the study of the interplay of regular and fractional diffusion. It is shown
that the main role of regular diffusion is to delay the onset of front
acceleration. In particular, the crossover time, $t_c$, to transition to the
accelerated fractional regime exhibits a logarithmic scaling of the form $t_c
sim log left(chi_d/chi_f
ight)$ where $chi_d$ and $chi_f$ are the
regular and fractional diffusivities. | | Source: | arXiv, 1409.3132 | | Services: | Forum | Review | PDF | Favorites | |
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