On the Accuracy of Some Absorbing Boundary Conditions for the Schrodinger Equation

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.

A Three-Level Parallelisation Scheme and Application to the Nelder-Mead Algorithm

“…First, we briefly present the problem which is used to test our methodology. We solve an initial-boundary value Schrödinger problem formulated in a finite space domain (Bugajev et al, 2017):…”

“…where ∂ n u is the normal derivative, ϕ k are solutions of the initial value problem for ODEs (Bugajev et al, 2017):…”

“…Here discrete approximations of PDEs represent the tasks v m in (1). To solve v m we must find solutions of N systems of linear equation with tridiagonal matrix (Bugajev et al, 2017). According to our three-level parallelisation scheme, the calculations of a single point in minimisation problem (11) define the block of tasks V k .…”