Domain Decomposition Method and High-Order Absorbing Boundary Conditions for the Numerical Simulation of the Time Dependent Schrödinger Equation with Ionization and Recombination by Intense Electric Field

• Published in: Journal of scientific computing Vol. 64; no. 3; pp. 620 - 646
• Main Authors: Antoine, X., Lorin, E., Bandrauk, A. D.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2015; Springer Nature B.V
• Subjects: Algorithms; Approximation; Boundary conditions; Computation; Computational Mathematics and Numerical Analysis; Decomposition; Domain decomposition methods; Electric fields; Electromagnetic fields; Electrons; Ionization; Lasers; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Nuclei; Schrodinger equation; Schroedinger equation; Theoretical; Time dependence; Velocity; Wave functions; Waveforms; Wavefunctions; Schwarz waveform relaxation; Ionization; Domain decomposition method; Schrödinger equation; Laser; Non-reflecting boundary conditions
• ISSN: 0885-7474; 1573-7691; 1573-7691
• DOI: 10.1007/s10915-014-9902-5

This paper is devoted to the efficient computation of the time dependent Schrödinger equation for quantum particles subject to intense electromagnetic fields including ionization and recombination of electrons with their parent ion. The proposed approach is based on a domain decomposition technique, allowing a fine computation of the wavefunction in the vicinity of the nuclei located in a domain Ω 1 and a fast computation in a roughly meshed domain Ω 2 far from the nuclei where the electrons are assumed free. The key ingredients in the method are (i) well designed transmission boundary conditions on ∂ Ω 1 (resp. ∂ Ω 2 ) in order to estimate the part of the wavefunction “leaving” Domain Ω 1 (resp. Ω 2 ), (ii) a Schwarz waveform relaxation algorithm to accurately reconstruct the solution. The developed method makes it possible for electrons to travel from one domain to another without loosing accuracy, when the frontier or the overlapping region between two domains is crossed by the wavefunction.