A lattice Boltzmann model for reaction dynamical
systems with time delay
Yu Xiaomeia, Shi Baochanga,b,*
aState Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, PR China
bDepartment of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, PR China
Nonlinear reaction dynamical systems with time delay are investigated to evaluate the capability of the lattice Boltzmann
model for delay differential dynamical systems. The stiff delay systems, multi-delay systems and two-dimensional partial
delay differential equation are studied. Computation precise, efficiency, delay appearances are observed. The simulation
results indicate that the lattice Boltzmann method is an innovative and effective numerical method to simulate the nonlinear
delay differential dynamical systems.
? 2006 Elsevier Inc. All rights reserved.
Keywords: Nonlinear reaction dynamical systems; Time delay; Lattice Boltzmann method
The lattice Boltzmann method (LBM) is an innovative numerical method based on kinetic theory to sim-
ulate various hydro-dynamical systems [1,2]. Although the LBM was developed only a decade ago, it has
shown large power to simulate various complex dynamical appearances [3–5]. Compared with conventional
numerical methods, it has advantages of simplicity, less computation, intrinsic parallel and so on. LBM
has been considered recently as a possible alternative explicit numerical scheme to conventional methods
for solving nonlinear macroscopic physical systems, in particular for Navier–Stokes flows. Consequently,
LBM has found applications in many areas of flow physics [6–8].
An important application of LBM is to solve a class of generalized reaction–diffusion systems, which
include ordinary differential dynamical systems, convection–diffusion, pure diffusion, Poisson equations. It
is more difficult than Navier–Stokes via LBM for the momentum flux is not conservative when reproduce
the desired form of nonlinear reaction–diffusion equations. Some studies using LBM to simulate nonlinear
reaction dynamical systems have been performed [9,10]. But seldom work has referred to the nonlinear
reaction–diffusion dynamical systems with time delay, which is more complex and can lead to surge and chaos.
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*Corresponding author. Address: Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074,
E-mail addresses: firstname.lastname@example.org (X. Yu), email@example.com (B. Shi).
Applied Mathematics and Computation 181 (2006) 958–965
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X. Yu, B. Shi / Applied Mathematics and Computation 181 (2006) 958–965