A Preconditioned Iterative Solver for the Scattering Solutions of the Schrödinger Equation

Abstract

Abstract. The Schrödinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schrödinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. We employ the iterative approach for their solution. In particular, we develop a preconditioner that has its spectrum restricted to a quadrant (of the complex plane) thereby making it easily invertible by multigrid methods with standard components. This multigrid preconditioner is used in conjuction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems. The aim of this study is to report the feasbility of this preconditioner for the model problems. We compare this idea with the other prevalent preconditioning ideas, and discuss its merits. Results of numerical experiments are presented, which complement the proposed ideas, and show that this preconditioner may be used in an automatic setting.

Solution of the Schrödinger equation using exterior complex scaling and fast Fourier transform

“…7(a) Finally let us make some remarks upon the convergence of the PCG approach used for E-SURFF. During the TFECS using with N = 1024 and preconditioner of the form (30) with N l = 10 the average (over all the calculated spectrum points) iterations number was N iter ≈ 79 until the discrepancy of the equation (22) was equal to ε = 10 −12 . In contrast, during the SECS using with the same grid and preconditioner parameters the average PCG iterations number was as small as N iter ≈ 36.…”

“…Since the matrix A is complex symmetric the system of equations (22) might be solved by means of an iteration method namely the preconditioned conjugated gradients (PCG) method [18,19]. On every PCG iteration one performs the multiplication of a vector by the matrix P −1 A, where P is a preconditioner matrix that satisfies the equation P −1 A ≈ I yet at the same time leads to a small operations number required for the equation Py = r solving.…”

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

“…The technical details of this transformation are given in [bin Zubair et al 2012]. Whenever we drop the p superscripts from here on, the symbols are generic for any of the channel PDEs.…”

“…With characteristics of the solver behaviour at hand, we finally study a realistic channel matrix for p = 2 idealised from the dynamics of Hydrogen or Deuterium [bin Zubair et al 2012]. This leads in the channels' frequency domain to two-dimensional Helmholtz problems…”

“…To the best of our knowledge, even results for a numerical solution with p ≥ 3 in our application area are rare. The aforementioned [Baertschy and Li 2001;Vanroose et al 2005;Horner et al 2007;bin Zubair et al 2012;Cools et al 2014a] for example all restrict to p ∈ {2, 3}, i.e. two-dimensional or three-dimensional grids.…”