Numerical simulation of a class of FitzHugh-Nagumo systems based on the lattice Boltzmann method

The lattice Boltz mann 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-infinity 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 u(0)(x) must satisfy vertical bar u(0)(x)vertical bar <= 1 and the parameters must satisfy theta i <= -tau(1 + alpha)Delta t/Delta 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.