A 5th-order method for 1D-device solution

The so-called Numerov process provides a three-point interpolation with an ~η
⁵
accuracy in grid's size η, much better than the standard finite-difference scheme that keeps the ~η
²
terms. Such a substantial improvement is achieved with a negligible increase in computational cost. As the method is applicable to second-order differential equations in one dimension, it is an ideal tool for solving, e.g., the Poisson and Schrödinger equations in ballistic electron devices, where the longitudinal (that is, along the channel) problem is typically separated from the lateral one and solved over a uniform grid. Despite its advantage, the Numerov process has found limited applications, due to the difficulty of keeping the same precision in the boundary conditions. A method to work out the boundary conditions consistently with the rest of the scheme is presented, and applications are shown.