A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schrödinger-Poisson equations with discontinuous potentials

Authors:
Tiao Lu;Wei Cai
Affiliations:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR China;Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
Venue:
Journal of Computational and Applied Mathematics
Year:
2008

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Abstract

In this paper, we propose a high order Fourier spectral-discontinuous Galerkin method for time-dependent Schrodinger-Poisson equations in 3-D spaces. The Fourier spectral Galerkin method is used for the two periodic transverse directions and a high order discontinuous Galerkin method for the longitudinal propagation direction. Such a combination results in a diagonal form for the differential operators along the transverse directions and a flexible method to handle the discontinuous potentials present in quantum heterojunction and supperlattice structures. As the derivative matrices are required for various time integration schemes such as the exponential time differencing and Crank Nicholson methods, explicit derivative matrices of the discontinuous Galerkin method of various orders are derived. Numerical results, using the proposed method with various time integration schemes, are provided to validate the method.