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 O(h(3/2)) in the L-2 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.