Domain decomposition algorithms for the two dimensional nonlinear Schrödinger equation and simulation of Bose–Einstein condensates
The SMAI journal of computational mathematics, Volume 2 (2016) , pp. 277-300.

In this paper, we apply the optimized Schwarz method to the two dimensional nonlinear Schrödinger equation and extend this method to the simulation of Bose–Einstein condensates (Gross– Pitaevskii equation). We propose an extended version of the Schwartz method by introducing a preconditioned algorithm. The two algorithms are studied numerically. The experiments show that the preconditioned algorithm improves the convergence rate and reduces the computation time. In addition, the classical Robin condition and a newly constructed absorbing condition are used as transmission conditions.

Published online:
DOI: https://doi.org/10.5802/smai-jcm.17
Classification: 35Q55,  65M55,  65Y05,  65M60
Keywords: nonlinear Schrödinger equation, rotating Bose–Einstein condensate, optimized Schwarz method, preconditioned algorithm, parallel algorithm
@article{SMAI-JCM_2016__2__277_0,
     author = {Christophe Besse and Feng Xing},
     title = {Domain decomposition algorithms for the two dimensional nonlinear {Schr\"odinger} equation and simulation of {Bose{\textendash}Einstein} condensates},
     journal = {The SMAI journal of computational mathematics},
     pages = {277--300},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {2},
     year = {2016},
     doi = {10.5802/smai-jcm.17},
     zbl = {1416.65328},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.17/}
}
Christophe Besse; Feng Xing. Domain decomposition algorithms for the two dimensional nonlinear Schrödinger equation and simulation of Bose–Einstein condensates. The SMAI journal of computational mathematics, Volume 2 (2016) , pp. 277-300. doi : 10.5802/smai-jcm.17. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.17/

[1] X. Antoine; W. Bao; C. Besse Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations, Comput. Phys. Commun., Volume 184 (2013) no. 12, pp. 2621-2633 | Article | Zbl 1344.35130

[2] X. Antoine; C. Besse; P. Klein Absorbing Boundary Conditions for the Two-Dimensional Schrödinger Equation With an Exterior Potential Part I: Construction and a Priori Estimates, Math. Model. Methods Appl. Sci., Volume 22 (2012) no. 10 | Article | Zbl 1251.35096

[3] X. Antoine; C. Besse; P. Klein Absorbing boundary conditions for the two-dimensional Schrödinger equation with an exterior potential. Part II: Discretization and numerical results, Numer. Math., Volume 125 (2013) no. 2, pp. 191-223 | Article | Zbl 06217651

[4] X. Antoine; R. Duboscq Computer Physics cations GPELab , a Matlab Toolbox to solve Gross–Pitaevskii Equations I : computation of stationary solutions, Comput. Phys. Commun., Volume 185 (2014) no. 11, pp. 2969-2991 | Article | Zbl 1348.35003

[5] X. Antoine; E. Lorin; A. D. Bandrauk 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, J. Sci. Comput. (2014), pp. 1-27 | Zbl 1332.78026

[6] S. Balay; M. F. Adams; J. Brown; P. Brune; K. Buschelman; V. Eijkhout; W. D. Gropp; D. Kaushik; M. G. Knepley; L. C. McInnes; K. Rupp; B. F. Smith; H. Zhang PETSc Users Manual (2013) no. ANL-95/11 - Revision 3.4 (Technical report)

[7] W. Bao; Y. Cai Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Model., Volume 6 (2012) no. 1, pp. 1-135 | Article | MR 3005624

[8] W. Bao; I.-L. Chern; F. Y. Lim Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose–Einstein condensates, J. Comput. Phys., Volume 219 (2006) no. 2, pp. 836-854 | Article | MR 2274959 | Zbl 1330.82031

[9] W. Bao; Q. Du Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., Volume 25 (2004) no. 5, pp. 1674-1697 | Article | MR 2087331 | Zbl 1061.82025

[10] W. Bao; D. Marahrens; Q. Tang; Y. Zhang A Simple and Efficient Numerical Method for Computing the Dynamics of Rotating Bose–Einstein Condensates via Rotating Lagrangian Coordinates, SIAM J. Sci. Comput., Volume 35 (2013) no. 6 | Article | MR 3129763 | Zbl 1286.35213

[11] W. Bao; P. A. Markowich; H. Wang Ground, Symmetric and Central Vortex States in Rotating Bose-Einstein Condensates, Commun. Math. Sci., Volume 3 (2005) no. 1, pp. 57-88 | Article | MR 2132826 | Zbl 1073.82004

[12] C. Besse; F. Xing Domain decomposition algorithms for two dimensional linear Schrödinger equation (2015) (https://arxiv.org/abs/1506.05639)

[13] C. Besse; F. Xing Schwarz waveform relaxation method for one dimensional Schrödinger equation with general potential, Numerical Algorithms, accepted (2016) | Article

[14] Y. Boubendir; X. Antoine; C. Geuzaine A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation, J. Comput. Phys., Volume 231 (2012) no. 2, pp. 262-280 | Article | MR 2872075 | Zbl 1243.65144

[15] A. Durán; J.-M. Sanz-Serna The numerical integration of relative equilibrium solutions. The nonlinear Schrodinger equation, IMA J. Numer. Anal., Volume 20 (2000) no. 2, pp. 235-261 | Article | MR 1752264 | Zbl 0954.65087

[16] M. J. Gander Optimized Schwarz Methods, SIAM J. Numer. Anal., Volume 44 (2006) no. 2, pp. 699-731 | Article | MR 2218966 | Zbl 1117.65165

[17] M. J. Gander Schwarz methods over the course of time, Electron. Trans. Numer. Anal., Volume 31 (2008), pp. 228-255 | MR 2569603 | Zbl 1171.65020

[18] M. J. Gander; L. Halpern Optimized Schwarz Waveform Relaxation Methods for Advection Reaction Diffusion Problems, SIAM J. Numer. Anal., Volume 45 (2007) no. 2, pp. 666-697 | Article | MR 2300292 | Zbl 1140.65063

[19] M. J. Gander; L. Halpern Méthodes de décomposition de domaine, Encyclopédie électronique pour les ingénieurs, 2012, 81 pages

[20] M. J. Gander; F. Magoules; F. Nataf Optimized Schwarz methods without overlap for the Helmholtz equation, SIAM J. Sci. Comput., Volume 24 (2002) no. 1, pp. 38-60 | Article | MR 1924414 | Zbl 1021.65061

[21] L. Halpern; J. Szeftel Optimized and quasi-optimal Schwarz waveform relaxation for the one dimensional Schrödinger equation, Math. Model. Methods Appl. Sci., Volume 20 (2010) no. 12, pp. 2167-2199 | Article | Zbl 1213.35192

[22] P.-L. Lions On the Schwarz alternating method. III: a variant for nonoverlapping subdomains, Third Int. Symp. domain Decompos. methods Partial Differ. equations, Volume 6 (1990), pp. 202-223 | Zbl 0704.65090

[23] S. Loisel Condition Number Estimates for the Nonoverlapping Optimized Schwarz Method and the 2-Lagrange Multiplier Method for General Domains and Cross Points, SIAM J. Numer. Anal., Volume 51 (2013) no. 6, pp. 3062-3083 | Article | MR 3129755 | Zbl 1287.35003

[24] Message Passing Interface Forum MPI : A Message-Passing Interface Standard Version 3.0, 2012 (https://spcl.inf.ethz.ch/Publications/index.php?pub=160)

[25] F. Nataf; F. Rogier Factorization of the convection-diffusion operator and a (possibly) non overlapping Schwarz method, Contemp. Math. (1994) | Article | Zbl 0796.65105