Lattice Boltzmann Method For The Complex Ginzburg-Landau Equation

Lattice Boltzmann method (LBM) is a numerical method for fluid flow and complex system which arises in the end of last century. It has some advantages that bring much attention, such as their algorithmic simplicity, parallel computation, easy handling of complex boundary conditions, and efficient numerical simulations. This method has been widely used to the nonlinear partial differential equation in recent years. In this paper, we will investigate the complex Ginzburg-Landau equation (CGLE) by using the LBM. The CGLE has the following form: where/β, a, d are all complex constants,▽2 is the Laplace operator. This is an amplitude equation, which governs complex variable. It describes the behavior of the system near a Hopf bifurcation. It has wide application in the fields of superconductivity, super fluidity and chemical system etc. However, we will focus on the dynamical evolution of the spiral wave and scroll wave.CGLE governs complex variable and has complex parameters; hence, we will build a lattice Boltzmann model suitable for complex variable.Under the assumption that the source term is second-order and by using the multi-scale technique, the Chapman-Enskog analysis on the lattice Boltzmann equation with complex distribution is given. Then, a series of partial differential equations on the complex equilibrium distribution are obtained. Under such condition, the FHP lattice in two dimensions (2D) and primitive cubic lattice in three dimensions (3D) are used to give the lattice Boltzmann model for the CGLE. The model has complex equilibrium distribution, complex additional distribution, complex lattice Boltzmann equation, and complex higher order moments of the equilibrium distribution. Meanwhile, an adjustable parameter is introduced in the additional distribution so that the model can be modified to give better results. The model has real single relaxation time factor, so the expression of the model has the same form as that for real partial differential equation. Thus, it retains the simplicity. In order to do the numerical simulation, the Neumann boundary condition is used. The equilibrium distributions on the boundaries are expressed by the macroscopic quantity. We assume that the distributions on the boundaries are equilibrium distributions. The error expressions for 2D and 3D are given separately because of the different higher order moments of discrete velocity. The accuracy of the model is the first order owing to the error rebounded phenomenon caused by the assumption that the source term is second order.First, the spiral wave dynamics is investigated in the numerical simulation. As to the results of the stable spiral wave, we find that there are two values of Im(a) which mark the change in the rotation and oscillation properties, respectively. And the spiral tips are a kind of topological point defects known as quantum vortices. At these defects,| A|= 0 theoretically, but the LBM result is around 2% to 8% . The profile of| A| going through the defects shows the linear increase near the defects, which is in agreement with theoretical prediction. Second, the LBM is used to investigate the unstable spiral wave under various conditions. With certain parameters, the single spiral wave will be broken up into small spirals and then the spiral turbulence state is formed finally. Before the spiral is broken up, we can observe that there is a low amplitude wave that spreads outwardly as the disk is growing. The target waves of amplitude can also be found between the low amplitude wave and the center.Compared with 2D problems, the computing costs of 3D problems are increasing sharply. Thus, we reduce the lattice size to 50×50×50. In order to test the validity of the 3D model, the spiral wave in quasi-two dimensions are simulated. The results show that such treatment is acceptable. Then, some initial conditions are given to form some specific stable scroll waves, such as scroll ring, helical scroll, multi-scroll waves and other scroll waves. The results of the motion of scroll ring are in agreement with classical results. The helical scroll has the similar law of motion to the scroll ring, but it will be a scroll wave with a straight filament finally. For the multi-scroll waves, the filaments are stretched and move until outside of the computing region in the evolution. Thus, the number of the filaments decreases as time goes on, and then they will all disappear. Such phenomenon is distinct from that of spiral waves in 2D CGLE. Additionally, two initial conditions for scroll wave with straight filaments are given. But the two have different destinations. The curved filament with its ends on the adjacent sides of the computing region will be an arc and move along the direction of its radius until it disappears. The filament with its ends on the opposite sides will be adjusted to a stable state, namely, a straight filament parallel to an axis. The results are in agreement with the theory of positive filament tension.The model in this paper gives acceptable results in the simulation of spiral wave and scroll wave. The model can be used to study other properties of spiral wave and scroll waves, and other complex partial differential equation. On the other hand, the real and imagery parts of the complex variable can be regarded as density of two components, thus the model can be extended to other bi-component reaction-diffusion systems.