High-order upwind finite volume element schemes for modelling of neuronal firing

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.

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