This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for linear equations with variable coefficients and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [X. Antoine and E. Lorin, Numer. Math. 137 (2017) 923–958] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.
Accepté le :
DOI : 10.1051/m2an/2017048
Mots-clés : Schwarz waveform relaxation, domain decomposition method, convergence rate, Schrödinger equation, Gross-Pitaevskii equation, absorbing boundary conditions, pseudodifferential operator theory, symbolical asymptotic expansion, stationary states, imaginary-time, normalized gradient flow
@article{M2AN_2018__52_4_1569_0,
author = {Antoine, Xavier and Hou, Fengji and Lorin, Emmanuel},
title = {Asymptotic estimates of the convergence of classical {Schwarz} waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1569--1596},
publisher = {EDP-Sciences},
volume = {52},
number = {4},
year = {2018},
doi = {10.1051/m2an/2017048},
zbl = {1407.65152},
mrnumber = {3878604},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2017048/}
}
TY - JOUR AU - Antoine, Xavier AU - Hou, Fengji AU - Lorin, Emmanuel TI - Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1569 EP - 1596 VL - 52 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2017048/ DO - 10.1051/m2an/2017048 LA - en ID - M2AN_2018__52_4_1569_0 ER -
%0 Journal Article %A Antoine, Xavier %A Hou, Fengji %A Lorin, Emmanuel %T Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1569-1596 %V 52 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2017048/ %R 10.1051/m2an/2017048 %G en %F M2AN_2018__52_4_1569_0
Antoine, Xavier; Hou, Fengji; Lorin, Emmanuel. Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1569-1596. doi : 10.1051/m2an/2017048. http://www.numdam.org/articles/10.1051/m2an/2017048/
[1] , and , Optimized waveform relaxation methods for longitudinal partitioning of transmission lines. IEEE Trans. Circuits Syst., 56 (2009) 1732–1743. | DOI | MR | Zbl
[2] , and , Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations. Comput. Phys. Commun. 184 (2013) 2621–2633. | DOI | MR | Zbl
[3] and , Construction, structure and asymptotic approximations of a microdifferential transparent boundary condition for the linear Schrödinger equation. J. Math. Pures Appl. 80 (2001) 701–738. | DOI | MR | Zbl
[4] , and , Artificial boundary conditions for one-dimensional cubic nonlinear Schrödinger equations. SIAM J. Numer. Anal. 43 (2006) 2272–2293. | DOI | MR | Zbl
[5] , and , Absorbing boundary conditions for the one-dimensional Schrödinger equation with an exterior repulsive potential. J. Comput. Phys. 228 (2009) 312–335. | DOI | MR | Zbl
[6] , and , Absorbing boundary conditions for the two-dimensional Schrödinger equation with an exterior potential. Part I: Construction and a priori estimates. Math. Models Methods Appl. Sci. 22 (2012) 1250026. | DOI | MR | Zbl
[7] , and , Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions. Math. Comput. 73 (2004) 1779–1799. | DOI | MR | Zbl
[8] , and , Towards accurate artificial boundary conditions for nonlinear PDEs through examples. Cubo 11 (2009) 29–48. | MR | Zbl
[9] and , GPELab, a Matlab toolbox to solve Gross-Pitaevskii equations I: Computation of stationary solutions. Comput. Phys. Commun. 185 (2014) 2969–2991. | DOI | Zbl
[10] and , Robust and efficient preconditioned Krylov spectral solvers for computing the ground states of fast rotating and strongly interacting Bose-Einstein condensates. J. Comput. Phys. 258C (2014) 509–523. | DOI | MR | Zbl
[11] and , Modeling and computation of Bose-Einstein condensates: stationary states, nucleation, dynamics, stochasticity, in Nonlinear Optical and Atomic Systems: at the Interface of Mathematics and Physics. CEMPI Subseries, 1st. In Vol. 2146 of Lect. Notes Math. Springer (2015) 49–145. | DOI | MR | Zbl
[12] and , An analysis of schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrödinger and Gross-Pitaevskii equations. Numer. Math. 137 (2017) 923–958. | DOI | MR | Zbl
[13] and , Lagrange Schwarz waveform relaxation domain decomposition methods for linear and nonlinear quantum wave problems. Appl. Math. Lett. 57 (2016) 38–45. | DOI | MR | Zbl
[14] , and , 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. 64 (2015) 620–646. | DOI | MR | Zbl
[15] , Ground states and dynamics of multicomponent Bose-Einstein condensates. Multiscale Model. Simul. 2 (2004) 210–236. | DOI | MR | Zbl
[16] and , Mathematical theory and numerical methods for Bose-Einstein condensation. Kinetic Relat. Models. 6 (2013) 1–135. | DOI | MR | Zbl
[17] , and , Efficient and spectrally accurate numerical methods for computing ground and first excited states in Bose-Einstein condensates. J. Comput. Phys. 219 (2006) 836–854. | DOI | MR | Zbl
[18] and , Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow. SIAM J. Sci. Comput. 25 (2004) 1674–1697. | DOI | MR | Zbl
[19] and , Ground-state solution of Bose-Einstein condensate by directly minimizing the energy functional. J. Comput. Phys. 187 (2003) 230–254. | DOI | MR | Zbl
[20] and , Resolution of the Gross-Pitaevskii equation with the imaginary-time method on a Lagrange mesh. Phys. Rev. E 82 (2010) 056701.
[21] and , Domain decomposition algorithms for two dimensional linear schrödinger equation. J. Sci. Comput. (2017) 1–26. | MR
[22] and , Schwarz waveform relaxation method for one-dimensional schrödinger equation with general potential. Numer. Algorithms 74 (2017) 393–426. | MR
[23] , and , Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm. Phys. Rev. E 62 (2000) 7438.
[24] , and , An introduction to domain decomposition methods: theory and parallel implementation (2015). | DOI | MR | Zbl
[25] and , Optimized Schwarz waveform relaxation methods for advection reaction diffusion problems. SIAM J. Num. Anal. 45 (2007). | DOI | MR | Zbl
[26] , Overlapping Schwarz for linear and nonlinear parabolic problems. Procs. of the 9th International Conference on Domain Decomposition (1996) 97–104.
[27] , Optimal Schwarz waveform relaxation methods for the one-dimensional wave equation. SIAM J. Numer. Anal. 41 (2003) 1643–1681. | DOI | MR | Zbl
[28] , Optimized Schwarz methods. SIAM J. Numer. Anal. 44 (2006) 699–731. | DOI | MR | Zbl
[29] , Optimized Schwarz waveform relaxation methods for advection diffusion problems. SIAM J. Numer. Anal. (2007) 666–697. | DOI | MR | Zbl
[30] , and , Optimal convergence for overlapping and non-overlapping Schwarz waveform relaxation. DDM.org, Augsburg (1999) 27–36. | MR
[31] , and , Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for parabolic problems. Preprint arXiv: (2015). | arXiv | MR
[32] and , Optimized and quasi-optimal Schwarz waveform relaxation for the one-dimensional Schrödinger equation. Math. Models Methods Appl. Sci. 20 (2010) 2167–2199. | DOI | MR | Zbl
[33] , Propagation des singularités des solutions d’équations pseudo-différentielles quasi homogènes. Ann. Inst. Fourier (Grenoble) 27 (1977) vii–viii, 79–123. | DOI | Numdam | MR | Zbl
[34] and , Non-homogeneous boundary value problems and applications. Vol. II, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182. Springer-Verlag, New York-Heidelberg (1972). | MR | Zbl
[35] , Schwarz waveform relaxation domain decomposition methodology for the N-body time-independent and time-dependent Schrödinger equation. Submitted (2017).
[36] , and , Frozen gaussian approximation based domain decomposition methods for the linear Schrödinger equation beyond the semi-classical regime. J. Comput. Phys. 315 (2016) 221–237. | DOI | MR | Zbl
[37] B., A time-dependent Dirichlet-Neumann method for the heat equation. Domain Decomposition Methods in Science and Engineering XXI. Springer International Publishing (2014) 467–475. | DOI | MR | Zbl
[38] , Lectures on linear partial differential equations. Amer. Math. Soc., Providence, R.I. (1973). | MR | Zbl
[39] and , Bose-Einstein condensation in dilute gases. Cambridge University Press (2002).
[40] and , Bose-Einstein condensation, vol. 116. Clarendon press (2003). | MR | Zbl
[41] and , Efficiently computing vortex lattices in rapid rotating Bose-Einstein condensates. Comput. Phys. Commun. 180 (2009) 854–860. | DOI | MR | Zbl
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