A fast and robust computational method for the ionization cross sections of the driven Schroedinger equation using an O(N) multigrid-based scheme

TL;DR

This paper introduces an efficient multigrid-based solver with a coupled channel correction step for the driven Schrödinger equation, achieving fast, scalable, and robust solutions across energy regimes.

Contribution

It develops a combined MG-CCCS iterative scheme that significantly improves convergence and scalability for complex Schrödinger problems, extending previous methods.

Findings

Achieves linear time complexity in the number of unknowns.

Demonstrates improved convergence rates over classical MG methods.

Validates optimal scalability on a 2D model problem.

Abstract

This paper improves the convergence and robustness of a multigrid-based solver for the cross sections of the driven Schroedinger equation. Adding an Coupled Channel Correction Step (CCCS) after each multigrid (MG) V-cycle efficiently removes the errors that remain after the V-cycle sweep. The combined iterative solution scheme (MG-CCCS) is shown to feature significantly improved convergence rates over the classical MG method at energies where bound states dominate the solution, resulting in a fast and scalable solution method for the complex-valued Schroedinger break-up problem for any energy regime. The proposed solver displays optimal scaling; a solution is found in a time that is linear in the number of unknowns. The method is validated on a 2D Temkin-Poet model problem, and convergence results both as a solver and preconditioner are provided to support the O(N) scalability of the method. This paper extends the applicability of the complex contour approach for far field map computation [S. Cools, B. Reps, W. Vanroose, An Efficient Multigrid Calculation of the Far Field Map for Helmholtz and Schroedinger Equations, SIAM J. Sci. Comp. 36(3) B367--B395, 2014].