A second-order, unconditionally invariant-set-preserving scheme for the FitzHugh-Nagumo equation

Abstract

In this paper, we present and analyze a second-order exponential time differencing Runge–Kutta (ETDRK2) scheme for the FitzHugh-Nagumo equation. Utilizing a second-order finite-difference space discretization, we derive the fully discrete numerical scheme by incorporating both the stabilization technique and the ETDRK2 scheme for temporal approximation. The smallest invariant set of the FitzHugh-Nagumo equation is presented. We demonstrate that the proposed scheme unconditionally preserves the invariant set without any time-step constraint. The convergence in both time and space is verified to achieve second-order accuracy. Numerical experiments are carried out to illustrate the efficiency, stability, and structure-preserving property of the proposed scheme.