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Abstract
In this paper, a semi-explicit scheme is constructed for the space-independent FitzHugh–Nagumo equation. Qualitative stability analysis shows that the semi-explicit scheme is dynamically consistent with the space independent equation. Then, the semi-explicit scheme is extended to construct a new finite difference scheme for the full FitzHugh–Nagumo equation in one- and two-space dimensions, respectively. According to the theory of M-matrices, it is proved that these new schemes are able to preserve the positivity and boundedness of solutions of the corresponding equations for arbitrary step sizes. The consistency and numerical stability of these schemes is also analysed. Combined with the property of the strictly diagonally dominant matrix, the convergence of these schemes is established. Numerical experiments illustrate our results and display the advantages of our schemes in comparison to some other schemes.
Acknowledgements
The authors wish to express their deepest gratitude to the anonymous referees and the editors for their valuable comments and suggestions, which have greatly improved this paper. This work was supported by NNSF of China (No. 11271101, No. 11401140), NNSF of Shandong Province of China (No. ZR2012AQ027), NSRIF in Harbin Institute of Technology (No. HIT. NSRIF. 2011095), SRF of Harbin Institute of Technology at Weihai (No. HIT (WH) XB201124).