Abstract
A detailed analysis of absorbing boundary conditions for the linear Schrodinger equation is presented in this paper. It is focused on absorbing boundary conditions that are obtained by using rational functions to approximate the exact transparent boundary conditions. Different strategies are investigated for the optimal selection of the coefficients of these rational functions, including the Pade approximation, the L2 norm approximations of the Fourier symbol, L2 minimization of a reflection coefficient, and two error minimization techniques for the chosen benchmark problems with known exact solutions. The results of computational experiments are given and a detailed comparison of the efficiency of these techniques is presented.
Highlights
In this paper, as a basic mathematical model, we consider the initial-value problem for the linear one-dimensional Schrodinger equation ∂u ∂2ui ∂t + ∂x2 =0, u(x, 0) = u0(x).−∞ < x < ∞, t > 0, (1.1)On the Accuracy of Some absorbing boundary conditions (ABCs) for the Schrodinger EquationThe Schrodinger equation is widely used for modelling quantum mechanics and non-linear optics problems
The equation is supplemented with an initial condition and only asymptotical behaviour of the solution at infinite boundaries is defined
We are interested to find a solution of problem (1.1) only on a finite domain so a proper restriction of the infinite domain is well suited for most real world modelling applications
Summary
As a basic mathematical model, we consider the initial-value problem for the linear one-dimensional Schrodinger equation. A good review on construction of transparent boundary conditions for differential problems and for discrete finite difference schemes, as well as stability analysis and computational experiments are given in [2, 9]. We mention papers where exact transparent boundary conditions are constructed and the stability of initial-boundary value problem is analysed for the differential Schrodinger equation [1, 2, 4]. These boundary conditions absorb the energy of waves as they reach the boundary area and attempt to minimize the amount of energy reflected by the artificial boundaries Different techniques, such as polynomials, splines or finite elements, can be used as a basis for approximations.
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