Domain Decomposition Algorithms for Two Dimensional Linear Schrödinger Equation

This paper deals with two domain decomposition methods for two dimensional linear Schrödinger equation, the Schwarz waveform relaxation method and the domain decomposition in space method. After presenting the classical algorithms, we propose a new algorithm for the Schrödinger equation with constant potential and a preconditioned algorithm for the general Schrödinger equation. These algorithms are then studied numerically. The numerical experiments show that the new algorithms can improve the convergence rate and reduce the computation time. Besides of the traditional Robin transmission condition, we also propose to use a newly constructed absorbing condition as the transmission condition.

[1]  Véronique Martin,et al.  An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions , 2004 .

[2]  Frédéric Nataf,et al.  FACTORIZATION OF THE CONVECTION-DIFFUSION OPERATOR AND THE SCHWARZ ALGORITHM , 1995 .

[3]  Martin J. Gander,et al.  Optimal Schwarz Waveform Relaxation for the One Dimensional Wave Equation , 2003, SIAM J. Numer. Anal..

[4]  Yanzhi Zhang,et al.  A Simple and Efficient Numerical Method for Computing the Dynamics of Rotating Bose-Einstein Condensates via Rotating Lagrangian Coordinates , 2013, SIAM J. Sci. Comput..

[5]  Jérémie Szeftel,et al.  Nonlinear nonoverlapping Schwarz waveform relaxation for semilinear wave propagation , 2007, Math. Comput..

[6]  Christophe Besse,et al.  A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations , 2008 .

[7]  Martin J. Gander,et al.  Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations , 2010, Networks Heterog. Media.

[8]  Christophe Geuzaine,et al.  A quasi-optimal domain decomposition algorithm for the time-harmonic Maxwell's equations , 2015, J. Comput. Phys..

[9]  David E. Keyes,et al.  Additive Schwarz Methods for Hyperbolic Equations , 1998 .

[10]  Martin J. Gander,et al.  Optimized Schwarz Waveform Relaxation Methods for Advection Reaction Diffusion Problems , 2007, SIAM J. Numer. Anal..

[11]  Martin J. Gander,et al.  Optimized Schwarz Methods , 2006, SIAM J. Numer. Anal..

[12]  Xiao-Chuan Cai,et al.  Multiplicative Schwarz Methods for Parabolic Problems , 1994, SIAM J. Sci. Comput..

[13]  Martin J. Gander,et al.  Schwarz Methods over the Course of Time , 2008 .

[14]  Christophe Besse,et al.  Communi-cations Computational methods for the dynamics of the nonlinear Schr̈odinger / Gross-Pitaevskii equations , 2013 .

[15]  Hua Xiang,et al.  A Coarse Space Construction Based on Local Dirichlet-to-Neumann Maps , 2011, SIAM J. Sci. Comput..

[16]  Anne Greenbaum,et al.  Any Nonincreasing Convergence Curve is Possible for GMRES , 1996, SIAM J. Matrix Anal. Appl..

[17]  Christophe Besse,et al.  Absorbing boundary conditions for the two-dimensional Schrödinger equation with an exterior potential , 2013, Numerische Mathematik.

[18]  Christian Cabos.  Error Bounds for Dynamic Responses in Forced Vibration Problems , 1994, SIAM J. Sci. Comput..

[19]  Christophe Geuzaine,et al.  GetDDM: An open framework for testing optimized Schwarz methods for time-harmonic wave problems , 2016, Comput. Phys. Commun..

[20]  Martin J. Gander,et al.  Optimized Schwarz Methods for Maxwell's Equations , 2006, SIAM J. Sci. Comput..

[21]  Jérôme Jaffré,et al.  Space-Time Domain Decomposition Methods for Diffusion Problems in Mixed Formulations , 2013, SIAM J. Numer. Anal..

[22]  Laurence Halpern,et al.  Optimized and Quasi-optimal Schwarz Waveform Relaxation for the One Dimensional Schrödinger Equation , 2010 .

[23]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[24]  Christophe Geuzaine,et al.  A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation , 2012, J. Comput. Phys..

[25]  Christophe Besse,et al.  Schwarz waveform relaxation method for one-dimensional Schrödinger equation with general potential , 2016, Numerical Algorithms.

[26]  Matthew G. Knepley,et al.  PETSc Users Manual (Rev. 3.4) , 2014 .

[27]  David Lannes,et al.  The Water Waves Problem: Mathematical Analysis and Asymptotics , 2013 .

[28]  Xavier Antoine,et al.  Domain decomposition methods and high-order absorbing boundary conditions for the numerical simulation of the time dependent Schrödinger equation with ionization and recombination by intense electric field , 2014 .

[29]  Christophe Besse,et al.  Absorbing Boundary Conditions for the Two-Dimensional Schrödinger Equation with an Exterior Potential. Part I: Construction and a priori Estimates , 2012 .