Article

# A lattice Boltzmann model for reaction dynamical systems with time delay

State Key Laboratory of Coal Combustion (SKLCC), Huazhong University of Science and Technology, Wu-han-shih, Hubei, China

Applied Mathematics and Computation (Impact Factor: 1.55). 10/2006; 181(2):958-965. DOI: 10.1016/j.amc.2006.02.020 Source: DBLP

**ABSTRACT**

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.

### Full-text preview

cscs.hust.edu.cn Available from: Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

- [Show abstract] [Hide abstract]

**ABSTRACT:**A lattice Boltzmann model for convection-diffusion equation with nonlinear convection and isotropic-diffusion terms is proposed through selecting equilibrium distribution function properly. The model can be applied to the common real and complex-valued nonlinear evolutionary equations, such as the nonlinear Schrödinger equation, complex Ginzburg-Landau equation, Burgers-Fisher equation, nonlinear heat conduction equation, and sine-Gordon equation, by using a real and complex-valued distribution function and relaxation time. Detailed simulations of these equations are performed, and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies. - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, a lattice Boltzmann model for one-dimensional nonlinear Dirac equation is presented by using double complex-valued distribution functions and carefully selected equilibrium distribution functions. The effects of space and time resolutions and relaxation time on the accuracy and stability of the model are numerically investigated in detail. It is found that the model is of second-order accuracy in both space and time, and the order of accuracy is near 3.0 at lower grid resolution, which shows that the lattice Boltzmann method is an effective numerical scheme for the nonlinear Dirac equation. - [Show abstract] [Hide abstract]

**ABSTRACT:**A general lattice Boltzmann (LB) model is proposed for solving nonlinear partial differential equations with the form $\partial_t \phi+\sum_{k=1}^{m} \alpha_k \partial_x^k \Pi_k (\phi)=0$, where $\alpha_k$ are constant coefficients, and $\Pi_k (\phi)$ are the known differential functions of $\phi$, $1\leq k\leq m \leq 6$. The model can be applied to the common nonlinear evolutionary equations, such as (m)KdV equation, KdV-Burgers equation, K($m,n$) equation, Kuramoto-Sivashinsky equation, and Kawahara equation, etc. Unlike the existing LB models, the correct constraints on moments of equilibrium distribution function in the proposed model are given by choosing suitable \emph{auxiliary-moments}, and how to exactly recover the macroscopic equations through Chapman-Enskog expansion is discussed in this paper. Detailed simulations of these equations are performed, and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies. Comment: 18 pages, 4 figures