格子玻尔兹曼方法在复杂的流体系统中得到了广泛的应用.本文针对在高于阈值常电流刺激下神经元动作电位周期性振荡的FitzHugh-Nagumo系统，构造了一类带源项和修正项的仿真格子玻尔兹曼模型.通过合理选择适当的局部平衡态分布函数和修正函数，再应用Chapman-Enskog多尺度分析，可以正确恢复出一类宏观非线性方程.通过积分法得到了修正函数的构造方法，并分析了格子玻尔兹曼模型L∞稳定的充分条件.利用网格相关性分析，本文所构造的模型具有二阶空间精度.应用本文所提出的模型，仿真模拟了几个具有解析解的初边值系统，并与传统的改进有限差分格式(MFDM)进行了对比，结果表明本文模型所得的数值解与解析解吻合，其模拟误差小于MFDM.此外，还针对不具有解析解的初边值系统进行了数值仿真，并与MFDM进行了对比.数值结果表明，两种计算格式的数值解比较吻合，进一步证明了本文所构造模型的有效性和稳定性.%The lattice Boltzmann method (LBM) was proposed as a novel mesoscopic numerical method, and is widely used to simulate complex nonlinear fluid systems. In this paper, we develop a lattice Boltzmann model with amending function and source term to solve a class of initial value problems of the FitzHugh Nagumo systems, which arises in the periodic oscillations of neuronal action potential under constant current stimulation higher than the threshold value. Firstly, we construct a non-standard lattice Boltzmann model with the proper amending function and source term. For different evolution equations, local equilibrium distribution functions and amending function are selected, and the nonlinear FitzHugh Nagumo systems can be recovered correctly by using the Chapman Enskog multi-scale analysis. Secondly, through the integral technique, we obtain a new method on how to construct the amending function. In order to guarantee the stability of the present model, the L∞ stability of the lattice Boltzmann model is analyzed by using the extremum principle, and we get a sufficient condition for the stability that is the initial value u0 (x) must satisfy |u0(x)| 6 1 and the parameters must satisfy θi 6 −τ(1+α)∆t∆x , (i = 1–4). Thirdly, based on the results of the grid independent analysis and numerical simulation, it can be concluded that the present model is convergent with two order space accuracy. Finally, some initial boundary value problems with analytical solutions are simulated to verify the effectiveness of the present model. The results are compared with the analytical solutions and numerical solutions obtained by the modified finite difference method (MFDM). It is shown that the numerical solutions agree well with the analytical solutions and the global relative errors obtained by the present model are smaller than the MFDM. Furthermore, some test problems without analytical solutions are numerically studied by the present model and the MFDM. The results show that the numerical solutions obtained by the present model are in good agreement with those obtained by the MFDM, which can validate the effectiveness and stability of the LBM. In conclusion, our model not only can enrich the applications of the lattice Boltzmann model in simulating nonlinear partial difference equations, but also help to provide valuable references for solving more complicated nonlinear partial difference systems. Therefore, this research has important theoretical significance and application value.
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