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- ItemNumerical algorithms for Schrödinger equation with artificial boundary conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Čiegis, Raimondas; Laukaitytė, Inga; Radziunas, MindaugasWe consider a one-dimensional linear Schrödinger problem defined on an infinite domain and approximated by the Crank-Nicolson type finite difference scheme. To solve this problem numerically we restrict the computational domain by introducing the reflective, absorbing or transparent artificial boundary conditions. We investigate the conservativity of the discrete scheme with respect to the mass and energy of the solution. Results of computational experiments are presented and the efficiency of different artificial boundary conditions is discussed.
- 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.