Coupling finite volume and nonstandard finite difference schemes for a singularly perturbed Schrödinger equation

摘要

The Schrödinger equation is a model for many physical processes in quantum physics. It isa singularly perturbed differential equation where the presence of the small reduced Planck’sconstant makes the classical numerical methods very costly and inefficient. We design twonew schemes. The first scheme is the nonstandard finite volume method, whereby the perturbationterm is approximated by nonstandard technique, the potential is approximated by itsmean value on the cell and the complex dependent boundary conditions are handled by exactschemes. In the second scheme, the deficiency of classical schemes is corrected by the innerexpansion in the boundary layer region. Numerical simulations supporting the performance ofthe schemes are presented.