格子玻尔兹曼方法研究激发介质中的非线性波
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摘要
激发介质中非线性波的性质是当前研究热点之一。在很多实际的系统中螺旋波及其破碎是有害的,因此对非线性波的性质研究有着重大的现实意义。随着格子玻尔兹曼方法的发展和成熟,并凭借其精度较高、程序代码简单、稳定性好等优点,成为了研究非线性波性质的有效工具。
本文采用格子玻尔兹曼方法,对Selkov反应扩散系统中的非线性波进行研究。首先建立了反应扩散方程的九速正方格子模型,由该模型的格子玻尔兹曼方程推导得到反应扩散方程。接着进行计算机数值模拟,模拟时将二维系统划分为300×300个格子,采用无流边界条件。主要的数值模拟工作由以下三部分组成:
一、选取分岔参数a =0.76, b =0.02,χ=0.1,κ=5.0, D X = DY=0.1,数值模拟结果显示系统具有可激发性;在相同参数下,从不同的初始态出发,系统可演化到达不同的非线性波;在不同的参数下,从相同的初始态出发,系统可演化到不同的三种状态:螺旋波、时空混沌态和均匀定态。发现Selkov反应扩散系统中螺旋波的失稳是Doppler失稳。对多种参数组合下的系统演化行为进行了模拟,得到了参数在一定范围内反映系统状态的相图。
二、根据格子玻尔兹曼方法理论定义了系统内能函数,其表达式为值模拟结果表明:在均匀态下激发介质中的内能随分岔参数a增大而线性增加,在螺旋波态下激发介质中的内能随参数a的增大反而以指数形式递减。在相同的系统参数下,系统的状态分别为行波、靶波、螺旋波时,系统内能随时间作小幅的周期变化,且变化幅度不相同,经处理后发现不同波态的平均内能相差不大,这是因为在参数相同时(即系统提供能量大致不变)由不同的初始条件得到不同的稳定波态,它们的内能应该比较接近。通过计算机数值计算螺旋波失稳前后系统的内能还发现:螺旋波失稳前后系统的内能急剧降低,这说明螺旋波失稳原因在于系统提供的能量不足。
三、根据九速正方格子模型,定义相应的格子玻尔兹曼熵函数H ,定义式为同参数不同波态的系统以及螺旋波失稳前后的熵进行计算机数值模拟,结果显示:相同参数下各稳定波态的熵随时间作小幅的周期变化,且变化幅度不相同,经处理发现不同稳定波态的熵并不相同,说明各稳定波态的有序度不相同。螺旋波失稳后系统的熵突然增大,这结果说明系统处于自发的相变过程时,系统总是向着熵增大的方向发展。
由于非线性系统的复杂性,激发介质中非线性波的性质还有许多问题有待深入研究,本文结束时在总结全文工作的基础上对此作了分析和展望。
The characteristic of nonlinear waves in excitable media is one of the current research focus. The spiral wave and its fragmentation usually do harm to the practical systems. Therefore, it has greatly practical significance to study the characteristic of the nonlinear wave. With many merits such as higher precision, simple code and excellent stability, Lattice Boltzmann method has become an effective tool for studying the nature of nonlinear wave.
In this paper, we studied the nonlinear wave in the Selkov reaction-diffusion system by using Lattice Boltzmann method. First of all, we established the 9-speed square lattice model for the reaction-diffusion equations and deduced the reaction-diffusion equations from the Lattice Boltzmann equation. Next, we divided the two-dimensional system into 300×300 grids and simulated it under the no-flow boundary condition by using computer. The main works of this paper include three parts as follow:
First, when taking the parameters as a =0.76, b =0.02,χ=0.1,κ=5.0, D_X = D_Y=0.1, the numerical simulation results show that the system is excitable. Under the same parameters, the system can evolve to different nonlinear waves starting from different initial states, and the system can evolve into three different states such as the spiral wave state, spatiotemporal chaos state and uniform state starting from the same state with different parameters. It is also found that the spiral wave instability in the Selkov reaction system is the Doppler instability. The system phase diagrams within a certain range of parameters are shown through the simulation of the system evolution under a variety of parameters.
Second, based on the theory of the lattice Boltzmann method, it is defined that the internal energy function is .Numerical simulation results show these as follows. In the homogeneous state, the internal energy of excitable medium increases linearly with the bifurcation parameter a increases, while in the spiral waves, the internal energy of excitable medium decreases in the index form with the bifurcation parameter a increases. Under the same parameter and the states of the system as traveling waves, target waves, spiral waves, respectively, internal energy of the system changes slightly and periodically with time. It is found that the mean internal energy in the system have little discrepancy, the reason is that the same parameters mean that the energy of the system is closed, even for the different stable states of waves. The calculation of internal energy of system before and after the spiral wave stability has been done, and the results show that internal energy of the system decreases rapidly when it loses stability, which indicates that spiral wave instability due to the poor support of the energy of the system.
Third, basing on the nine-velocity grid model, it is defined that an entropy function according to the Lattice Boltzmann method as Simulations of the entropy of the different wave states with the same parameters have been done, and the entropy of the spiral wave before and after it loses instability is shown in this text, too. Under the same parameters,the entropy of different stable wave states changes lightly and periodically with time. After the treatment, it is found that entropy is not the same for different wave states, which indicates that the order degree of each wave states is not the same. When spiral wave became instability, the entropy increases suddenly, which indicates that the system evolve to the state whose entropy is bigger when the system undergoing the process of spontaneous phase transition.
As the complexity of nonlinear systems, many issues about the nature of nonlinear waves in excitable media need further research. At the end of this paper, the summary and outlook are shown.
本文采用格子玻尔兹曼方法,对Selkov反应扩散系统中的非线性波进行研究。首先建立了反应扩散方程的九速正方格子模型,由该模型的格子玻尔兹曼方程推导得到反应扩散方程。接着进行计算机数值模拟,模拟时将二维系统划分为300×300个格子,采用无流边界条件。主要的数值模拟工作由以下三部分组成:
一、选取分岔参数a =0.76, b =0.02,χ=0.1,κ=5.0, D X = DY=0.1,数值模拟结果显示系统具有可激发性;在相同参数下,从不同的初始态出发,系统可演化到达不同的非线性波;在不同的参数下,从相同的初始态出发,系统可演化到不同的三种状态:螺旋波、时空混沌态和均匀定态。发现Selkov反应扩散系统中螺旋波的失稳是Doppler失稳。对多种参数组合下的系统演化行为进行了模拟,得到了参数在一定范围内反映系统状态的相图。
二、根据格子玻尔兹曼方法理论定义了系统内能函数,其表达式为值模拟结果表明:在均匀态下激发介质中的内能随分岔参数a增大而线性增加,在螺旋波态下激发介质中的内能随参数a的增大反而以指数形式递减。在相同的系统参数下,系统的状态分别为行波、靶波、螺旋波时,系统内能随时间作小幅的周期变化,且变化幅度不相同,经处理后发现不同波态的平均内能相差不大,这是因为在参数相同时(即系统提供能量大致不变)由不同的初始条件得到不同的稳定波态,它们的内能应该比较接近。通过计算机数值计算螺旋波失稳前后系统的内能还发现:螺旋波失稳前后系统的内能急剧降低,这说明螺旋波失稳原因在于系统提供的能量不足。
三、根据九速正方格子模型,定义相应的格子玻尔兹曼熵函数H ,定义式为同参数不同波态的系统以及螺旋波失稳前后的熵进行计算机数值模拟,结果显示:相同参数下各稳定波态的熵随时间作小幅的周期变化,且变化幅度不相同,经处理发现不同稳定波态的熵并不相同,说明各稳定波态的有序度不相同。螺旋波失稳后系统的熵突然增大,这结果说明系统处于自发的相变过程时,系统总是向着熵增大的方向发展。
由于非线性系统的复杂性,激发介质中非线性波的性质还有许多问题有待深入研究,本文结束时在总结全文工作的基础上对此作了分析和展望。
The characteristic of nonlinear waves in excitable media is one of the current research focus. The spiral wave and its fragmentation usually do harm to the practical systems. Therefore, it has greatly practical significance to study the characteristic of the nonlinear wave. With many merits such as higher precision, simple code and excellent stability, Lattice Boltzmann method has become an effective tool for studying the nature of nonlinear wave.
In this paper, we studied the nonlinear wave in the Selkov reaction-diffusion system by using Lattice Boltzmann method. First of all, we established the 9-speed square lattice model for the reaction-diffusion equations and deduced the reaction-diffusion equations from the Lattice Boltzmann equation. Next, we divided the two-dimensional system into 300×300 grids and simulated it under the no-flow boundary condition by using computer. The main works of this paper include three parts as follow:
First, when taking the parameters as a =0.76, b =0.02,χ=0.1,κ=5.0, D_X = D_Y=0.1, the numerical simulation results show that the system is excitable. Under the same parameters, the system can evolve to different nonlinear waves starting from different initial states, and the system can evolve into three different states such as the spiral wave state, spatiotemporal chaos state and uniform state starting from the same state with different parameters. It is also found that the spiral wave instability in the Selkov reaction system is the Doppler instability. The system phase diagrams within a certain range of parameters are shown through the simulation of the system evolution under a variety of parameters.
Second, based on the theory of the lattice Boltzmann method, it is defined that the internal energy function is .Numerical simulation results show these as follows. In the homogeneous state, the internal energy of excitable medium increases linearly with the bifurcation parameter a increases, while in the spiral waves, the internal energy of excitable medium decreases in the index form with the bifurcation parameter a increases. Under the same parameter and the states of the system as traveling waves, target waves, spiral waves, respectively, internal energy of the system changes slightly and periodically with time. It is found that the mean internal energy in the system have little discrepancy, the reason is that the same parameters mean that the energy of the system is closed, even for the different stable states of waves. The calculation of internal energy of system before and after the spiral wave stability has been done, and the results show that internal energy of the system decreases rapidly when it loses stability, which indicates that spiral wave instability due to the poor support of the energy of the system.
Third, basing on the nine-velocity grid model, it is defined that an entropy function according to the Lattice Boltzmann method as Simulations of the entropy of the different wave states with the same parameters have been done, and the entropy of the spiral wave before and after it loses instability is shown in this text, too. Under the same parameters,the entropy of different stable wave states changes lightly and periodically with time. After the treatment, it is found that entropy is not the same for different wave states, which indicates that the order degree of each wave states is not the same. When spiral wave became instability, the entropy increases suddenly, which indicates that the system evolve to the state whose entropy is bigger when the system undergoing the process of spontaneous phase transition.
As the complexity of nonlinear systems, many issues about the nature of nonlinear waves in excitable media need further research. At the end of this paper, the summary and outlook are shown.
引文
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