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Abstract
In this paper, we design high-order upwind finite volume element method (UFVEM) schemes to solve first-order hyperbolic partial differential-difference equation with shift, which arises in the modelling of neuronal firing. Error analysis for the schemes shows that the approximate solution obtained from UFVEM schemes is in the L2 norm. Numerical examples are also provided to support the method and the theoretical analysis. The new schemes preserve the height of neuronal impulses better than usual Lax–Friedrichs schemes which we have shown in numerical examples. In addition, the numerical oscillation produced by implicit centre finite difference schemes can also be eliminated.
Acknowledgements
This project is partially supported by the National Natural Science Foundation of China grant no. 11071123, the Innovation Program for University Postgraduates in Jiangsu Province nos. CXZZ12_0382 and CXLX12_0388, and Scientific Research Fund of Zhejiang Provincial Education Department no. Y201327415. The authors would like to thank the editor and the anonymous referees for their invaluable comments and suggestions which have helped to improve the paper greatly.