A finite-difference scheme to approximate non-negative and bounded solutions of a FitzHugh–Nagumo equation

Abstract

In this work, we present a finite-difference scheme that preserves the non-negativity and the boundedness of some solutions of a FitzHugh–Nagumo equation. The method is explicit, and it approximates the solutions of the nonlinear, parabolic partial differential equation under study with a consistency of order 𝒪 (Δ t+(Δ x)2) in the Dirichlet regime investigated. We give sufficient conditions in terms of the computational and the model parameters, in order to guarantee the non-negativity and the boundedness of the approximations. We also provide analyses of consistency, linear stability and convergence of the method. Our simulations establish that the properties of non-negativity and boundedness are actually preserved by the scheme when the proposed constraints are satisfied. Finally, a comparison against some second-order accurate methods reveals that our technique is easier to implement computationally, and it is better at preserving the properties of non-negativity and boundedness of the solutions of the FitzHugh–Nagumo equation under study.