用于复Ginzburg-Landau方程的格子Boltzmann方法
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摘要
格子Boltzmann方法是上世纪末兴起的一种用于流体力学模拟及复杂系统计算的数值方法。它具有算法简单有效、计算并行、复杂边界容易处理等优点,因此受到了广泛关注。近几年,格子Boltzmann方法广泛应用于线性及非线性偏微分方程的求解及模拟。本文用此方法研究复Ginzburg-Landau方程。方程如下：其中β,a,d都是复常数,(?)2是Laplace算子。这是一个具有复变量的振幅方程,描述的是Hopf分叉附近的系统的行为。它在超导、超流、化学系统等领域中,都有广泛的应用。本文重点研究螺旋波及三维涡卷的动力学演化。
复Ginzburg-Landau方程具有复变量和复方程参数,因此本文构建直接用于复变量的格子Boltzmann模型。
在源项的二阶假设下,使用多尺度技术,通过对具有源项及复分布函数的格子Boltzmann方程做Chapman分析,得到一系列关于复平衡态分布函数的偏微分方程。采用二维FHP格子和三维主立方体格子,给出了用于二维及三维复Ginzburg-Landau方程的格子Boltzmann模型。此模型具有复平衡态分布、复附加分布、复格子Boltzmann方程及平衡态分布的复高阶矩形式。同时,针对附加分布函数,我们引入了一个可调参数,以便对模型做出适当调整来改善结果。模型具有实数的单松弛时间因子,因此模型的表达与用于实偏微分方程的模型形式一致。这使得模型保留了用于实方程的模型的简单性。为便于数值模拟,采用Neumann边界条件。边界上的平衡态分布函数由宏观边界的表达式给出。假定在边界处的分布函数为平衡态分布。因为二维及三维的高阶离散速度矩表达式不同,本文分别给出其误差表达式。由于二阶假设导致误差反弹现象,使得模型具有一阶精度。
在本文中,首先研究螺旋波的动力学演化。在对稳定螺旋波的研究中,发现Im(a)有两个值分别标志着其旋转和振动性质的改变。另外,螺旋端点是一种拓扑缺陷点,称为量子涡。在这些点处,理论上|A|=0,但格子Boltzmann模型的结果在2%到8%之间。穿过量子涡剖面的|A|在缺陷点附近线性增加,这与理论预测一致。第二,研究了不同条件下不稳定螺旋波的失稳性质。在特定参数条件下,形成的单螺旋波会破碎成众多的不规则小螺旋,系统进入螺旋湍流态。在螺旋破碎之前,有低振幅波向外传播,在计算区域中心和低振幅波之间存在着振幅的靶波。
与二维问题相比,三维问题的计算消耗明显增加。因此,为研究涡卷性质,将网格调整为50×50×50。为测试三维模型的有效性,先用其模拟准二维空间中的螺旋波。结果表明,当前的网格尺寸可以用来研究三维涡卷的某些性质。然后,在不同的初始条件下,分别给出了各种稳定涡卷的演化,如涡环、涡线为螺旋线的涡卷、不规则多个涡卷和其他类型的涡卷。涡环运动的结果与经典结果一致。螺旋状涡线的涡卷与涡环的运动相似,但最后会形成直涡线的稳定涡卷。对于随机给定的初始条件形成的涡卷,其涡线形成时一般均为弯曲状,经过相互作用,涡线将因拉伸而运动出计算区域以外。因此,在演化中涡线数目逐渐减少,与二维情形不同,最终涡线将全部消失。另外,对于其他两种直线型涡卷,当涡线两端位于计算区域两相邻侧面时,涡线将弯成圆弧,然后沿着半径方向运动至消失；而当涡线两端位于两相对侧面时,将演化成直线型涡卷。因此,涡卷的演化结果为或者随着演化消失,全场变成均匀态,或者经过拉伸变形后形成与坐标轴平行的直涡线的涡卷,这符合正涡线拉伸理论。
对二维螺旋波及三维涡卷的模拟和研究,格子Boltzmann模型都给出了可以接受的结果。它可以用来研究螺旋波或涡卷的其他性质和其他的复偏微分方程。由于复变量的实虚部可以分别看作是两种物质的浓度,因此可以将此模型拓展到其他的双组分反应扩散系统的研究中。
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.
复Ginzburg-Landau方程具有复变量和复方程参数,因此本文构建直接用于复变量的格子Boltzmann模型。
在源项的二阶假设下,使用多尺度技术,通过对具有源项及复分布函数的格子Boltzmann方程做Chapman分析,得到一系列关于复平衡态分布函数的偏微分方程。采用二维FHP格子和三维主立方体格子,给出了用于二维及三维复Ginzburg-Landau方程的格子Boltzmann模型。此模型具有复平衡态分布、复附加分布、复格子Boltzmann方程及平衡态分布的复高阶矩形式。同时,针对附加分布函数,我们引入了一个可调参数,以便对模型做出适当调整来改善结果。模型具有实数的单松弛时间因子,因此模型的表达与用于实偏微分方程的模型形式一致。这使得模型保留了用于实方程的模型的简单性。为便于数值模拟,采用Neumann边界条件。边界上的平衡态分布函数由宏观边界的表达式给出。假定在边界处的分布函数为平衡态分布。因为二维及三维的高阶离散速度矩表达式不同,本文分别给出其误差表达式。由于二阶假设导致误差反弹现象,使得模型具有一阶精度。
在本文中,首先研究螺旋波的动力学演化。在对稳定螺旋波的研究中,发现Im(a)有两个值分别标志着其旋转和振动性质的改变。另外,螺旋端点是一种拓扑缺陷点,称为量子涡。在这些点处,理论上|A|=0,但格子Boltzmann模型的结果在2%到8%之间。穿过量子涡剖面的|A|在缺陷点附近线性增加,这与理论预测一致。第二,研究了不同条件下不稳定螺旋波的失稳性质。在特定参数条件下,形成的单螺旋波会破碎成众多的不规则小螺旋,系统进入螺旋湍流态。在螺旋破碎之前,有低振幅波向外传播,在计算区域中心和低振幅波之间存在着振幅的靶波。
与二维问题相比,三维问题的计算消耗明显增加。因此,为研究涡卷性质,将网格调整为50×50×50。为测试三维模型的有效性,先用其模拟准二维空间中的螺旋波。结果表明,当前的网格尺寸可以用来研究三维涡卷的某些性质。然后,在不同的初始条件下,分别给出了各种稳定涡卷的演化,如涡环、涡线为螺旋线的涡卷、不规则多个涡卷和其他类型的涡卷。涡环运动的结果与经典结果一致。螺旋状涡线的涡卷与涡环的运动相似,但最后会形成直涡线的稳定涡卷。对于随机给定的初始条件形成的涡卷,其涡线形成时一般均为弯曲状,经过相互作用,涡线将因拉伸而运动出计算区域以外。因此,在演化中涡线数目逐渐减少,与二维情形不同,最终涡线将全部消失。另外,对于其他两种直线型涡卷,当涡线两端位于计算区域两相邻侧面时,涡线将弯成圆弧,然后沿着半径方向运动至消失；而当涡线两端位于两相对侧面时,将演化成直线型涡卷。因此,涡卷的演化结果为或者随着演化消失,全场变成均匀态,或者经过拉伸变形后形成与坐标轴平行的直涡线的涡卷,这符合正涡线拉伸理论。
对二维螺旋波及三维涡卷的模拟和研究,格子Boltzmann模型都给出了可以接受的结果。它可以用来研究螺旋波或涡卷的其他性质和其他的复偏微分方程。由于复变量的实虚部可以分别看作是两种物质的浓度,因此可以将此模型拓展到其他的双组分反应扩散系统的研究中。
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.
引文
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