Abstract:
The Schrödinger equation is a model for many physical processes in quantum physics. It is
a singularly perturbed differential equation where the presence of the small reduced Planck’s
constant makes the classical numerical methods very costly and inefficient. We design two
new schemes. The first scheme is the nonstandard finite volume method, whereby the perturbation
term is approximated by nonstandard technique, the potential is approximated by its
mean value on the cell and the complex dependent boundary conditions are handled by exact
schemes. In the second scheme, the deficiency of classical schemes is corrected by the inner
expansion in the boundary layer region. Numerical simulations supporting the performance of
the schemes are presented.
