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

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.

Keywords:

• Schrödinger equation
• oscillatory preserving schemes
• finite volume method
• nonstandard finite difference method

2010 AMS Subject Classifications:

• 65M06
• 65M08
• 97N40

Acknowledgments

Thanks are addressed to the four anonymous reviewers whose suggestions have contributed to the improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors acknowledge the support of South African NRF and DST/NRF SARChI Chair on Mathematical Models and Methods in Bioengineering and Biosciences (M${}^3$B${}^2$).