This paper is devoted to the numerical approximation of the spatially extended FitzHugh–Nagumo transport equation with strong local interactions based on a particle method. In this regime, the time step can be subject to stability constraints related to the interaction kernel. To avoid this limitation, our approach is based on higher-order implicit-explicit numerical schemes. Thus, when the magnitude of the interactions becomes large, this method provides a consistent discretization of the macroscopic reaction-diffusion FitzHugh–Nagumo system. We carry out some theoretical proofs and perform several numerical experiments that establish a solid validation of the method and its underlying concepts.

本文致力于基于粒子方法的具有强局部相互作用的空间扩展的FitzHugh-Nagumo输运方程的数值逼近。在这种情况下，时间步长可能会受到与交互内核相关的稳定性约束的影响。为了避免这种限制，我们的方法基于高阶隐式-显式数值方案。因此，当相互作用的大小变大时，此方法将使宏观反应扩散FitzHugh-Nagumo系统的离散化保持一致。我们进行了一些理论上的证明，并进行了一些数值实验，建立了对该方法及其基础概念的可靠验证。