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- ItemNumerical simulation of quantum waveguides(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Arnold, Anton; Ehrhardt, Matthias; Schulte, MaikeThis chapter is a review of the research of the authors from the last decade and focuses on the mathematical analysis of the Schrödinger model for nano-scale semiconductor devices. We discuss transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation on a two dimensional domain. First we derive the two dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson-type finite difference scheme and a compact nine-point scheme. For this difference equations we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems. The presented discrete boundary-valued problem is unconditionally stable and completely reflection-free at the boundary. Then, since the DTBCs for the Schrödinger equation include a convolution w.r.t. time with a weakly decaying kernel, we construct approximate DTBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. In several numerical tests we illustrate the perfect absorption of outgoing waves independent of their impact angle at the boundary, the stability, and efficiency of the proposed method. Finally, we apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron inflow.
- ItemDiscrete transparent boundary conditions for the Schrödinger equation on circular domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Arnold, Anton; Ehrhardt, Matthias; Schulte, Maike; Sofronov, IvanWe propose transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation on a circular computational domain. First we derive the two-dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson finite difference scheme. The presented discrete initial boundary-value problem is unconditionally stable and completely reflection-free at the boundary. Then, since the discrete TBCs for the Schrödinger equation with a spatially dependent potential include a convolution w.r.t. time with a weakly decaying kernel, we construct approximate discrete TBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. In numerical tests we finally illustrate the accuracy, stability, and efficiency of the proposed method. As a by-product we also present a new formulation of discrete TBCs for the 1D Schrödinger equation, with convolution coefficients that have better decay properties than those from the literature.
- ItemA review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Antoine, Xavier; Arnold, Anton; Besse, Chritophe; Ehrhardt, Matthias; Schädle, AchimIn this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.