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High-order IMEX-spectral schemes for computing the dynamics of systems of nonlinear Schrödinger/Gross-Pitaevskii equations. (English) Zbl 1422.65277
Summary: The aim of this paper is to build and validate some explicit high-order schemes, both in space and time, for simulating the dynamics of systems of nonlinear Schrödinger/Gross-Pitaevskii equations. The method is based on the combination of high-order IMplicit-EXplicit (IMEX) schemes in time and Fourier pseudo-spectral approximations in space. The resulting IMEXSP schemes are highly accurate, efficient and easy to implement. They are also robust when used in conjunction with an adaptive time stepping strategy and appear as an interesting alternative to time-splitting pseudo-spectral (TSSP) schemes. Finally, a complete numerical study is developed to investigate the properties of the IMEXSP schemes, in comparison with TSSP schemes, for one- and two-components systems of Gross-Pitaevskii equations.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
Software:
GPELab
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[1] Aftalion, A.; Mason, P., Phase diagrams and Thomas-Fermi estimates for spin-orbit-coupled Bose-Einstein condensates under rotation, Phys. Rev. A, 88, (Aug. 2013)
[2] Anderson, M. H.; Ensher, J. R.; Matthews, M. R.; Wieman, C. E.; Cornell, E. A., Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269, 5221, 198-201, (Jul. 14, 1995)
[3] Antoine, X.; Bao, W.; Besse, C., Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., 184, 12, 2621-2633, (2013) · Zbl 1344.35130
[4] Antoine, X.; Duboscq, R., Gpelab, a Matlab toolbox to solve Gross-Pitaevskii equations I: computation of stationary solutions, Comput. Phys. Commun., 185, 11, 2969-2991, (2014) · Zbl 1348.35003
[5] Antoine, X.; Duboscq, R., Robust and efficient preconditioned Krylov spectral solvers for computing the ground states of fast rotating and strongly interacting Bose-Einstein condensates, J. Comput. Phys., 258, 509-523, (2014) · Zbl 1349.82027
[6] Antoine, X.; Duboscq, R., Gpelab, a Matlab toolbox to solve Gross-Pitaevskii equations II: dynamics and stochastic simulations, Comput. Phys. Commun., 193, 95-117, (2015) · Zbl 1344.82004
[7] Antoine, X.; Duboscq, R., Modeling and computation of Bose-Einstein condensates: stationary states, nucleation, dynamics, stochasticity, (Besse, C.; Garreau, J. C., Nonlinear Optical and Atomic Systems: At the Interface of Physics and Mathematics, Lecture Notes in Mathematics, vol. 2146, (2015)), 49-145 · Zbl 1344.35114
[8] Antonelli, P.; Marahrens, D.; Sparber, C., On the Cauchy problem for nonlinear Schrödinger equations with rotation, Discrete Contin. Dyn. Syst., 32, 3, 703-715, (2012) · Zbl 1234.35238
[9] Ascher, U. M.; Ruuth, S. J.; Spiteri, R. J., Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Special Issue on Time Integration, Amsterdam, 1996, Appl. Numer. Math., 25, 2-3, 151-167, (1997) · Zbl 0896.65061
[10] Bao, W.; Cai, Y., Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6, 1, 1-135, (2013) · Zbl 1266.82009
[11] Bao, W.; Marahrens, D.; Tang, Q.; Zhang, Y., A simple and efficient numerical method for computing the dynamics of rotating Bose-Einstein condensates via rotating Lagrangian coordinates, SIAM J. Sci. Comput., 35, 6, (2013) · Zbl 1286.35213
[12] Bao, W.; Wang, H., An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose-Einstein condensates, J. Comput. Phys., 217, 2, 612-626, (2006) · Zbl 1160.82343
[13] Belmonte-Beitia, J.; Cuevas, J., Solitons for the cubic-quintic nonlinear Schrödinger equation with time- and space-modulated coefficients, J. Phys. A, 42, 16, (2009) · Zbl 1167.35041
[14] Besse, C.; Dujardin, G.; Lacroix-Violet, I., High order exponential integrators for nonlinear Schrödinger equations with application to rotating Bose-Einstein condensates, (2015), preprint · Zbl 1371.35235
[15] Blanes, S.; Casas, F., Splitting methods for non-autonomous separable dynamical system, J. Phys. A, Math. Gen., 39, 5405-5423, (2006) · Zbl 1090.65083
[16] Blanes, S.; Casas, F.; Murua, A., Splitting methods for non-autonomous linear systems, Int. J. Comput. Math., 6, 713-727, (2007) · Zbl 1121.65081
[17] Blanes, S.; Casas, F.; Murua, A., Splitting methods in the numerical integration of non-autonomous dynamical systems, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., 106, 1, 49-66, (2012) · Zbl 1259.65109
[18] Blanes, S.; Casas, F.; Murua, A., An efficient algorithm based on splitting for the time integration of the Schrödinger equation, J. Comput. Phys., 303, 396-412, (2015) · Zbl 1349.65393
[19] Blanes, S.; Moan, P. C., Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods, J. Comput. Appl. Math., 142, 2, 313-330, (2002) · Zbl 1001.65078
[20] Blanes, S.; Moan, P. C., Splitting methods for non-autonomous differential equations, J. Comput. Phys., 170, 205-230, (2001) · Zbl 0986.65125
[21] Butcher, J. C., The numerical analysis of ordinary differential equations, (1987), Wiley Chichester · Zbl 0616.65072
[22] Butcher, J. C., A history of Runge-Kutta methods, Selected Keynote Papers Presented at 14th IMACS World Congress, Atlanta, GA, 1994, Appl. Numer. Math., 20, 3, 247-260, (1996) · Zbl 0858.65073
[23] Carpenter, M. H.; Kennedy, C. A., Fourth-order 2N-storage Runge-Kutta schemes, NASA Tech. Memo., 109112, 871-885, (1994)
[24] Castella, F.; Chartier, P.; Descombes, S.; Vilmart, G., Splitting methods with complex times for parabolic equations, BIT Numer. Math., 49, 3, 487-508, (2009) · Zbl 1180.65106
[25] Descombes, S.; Duarte, M.; Dumont, T.; Louvet, V.; Massot, M., Adaptive time splitting method for multi-scale evolutionary partial differential equations, Confluentes Math., 3, 3, 413-443, (2011) · Zbl 1231.65153
[26] Dimarco, G.; Mieussens, L.; Rispoli, V., An asymptotic preserving automatic domain decomposition method for the Vlasov-Poisson-BGK system with applications to plasmas, J. Comput. Phys., 274, 122-139, (2014) · Zbl 1351.76101
[27] Dimarco, G.; Pareschi, L.; Rispoli, V., Implicit-explicit Runge-Kutta schemes for the Boltzmann-Poisson system for semiconductors, Commun. Comput. Phys., 15, 5, 1291-1319, (2014) · Zbl 1373.82074
[28] Fetter, A. L., Vortex dynamics in spin-orbit-coupled Bose-Einstein condensates, Phys. Rev. A, 89, 2, (2014)
[29] Hairer, E.; Lubich, C.; Wanner, G., Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, vol. 31, (2006), Springer Science & Business Media · Zbl 1094.65125
[30] Kasamatsu, K., Dynamics of quantized vortices in Bose-Einstein condensates with laser-induced spin-orbit coupling, Phys. Rev. A, 92, 6, (2015)
[31] Kennedy, C. A.; Carpenter, M. H., Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44, 1, 139-181, (January 2003)
[32] Liu, H.; Zou, J., Some new additive Runge-Kutta methods and their applications, J. Comput. Appl. Math., 190, 1-2, 74-98, (2006) · Zbl 1089.65066
[33] Ma, W. X.; Chen, M., Direct search for exact solutions to the nonlinear Schrödinger equation, Appl. Math. Comput., 215, 8, 2835-2842, (2009) · Zbl 1180.65130
[34] Mason, P.; Aftalion, A., Classification of the ground states and topological defects in a rotating two-component Bose-Einstein condensate, Phys. Rev. A, 84, 3, (2011)
[35] Ming, J.; Tang, Q.; Zhang, Y., An efficient spectral method for computing dynamics of rotating two-components Bose-Einstein condensates via coordinate transformation, J. Comput. Phys., 258, 538-554, (2014) · Zbl 1349.82102
[36] Qu, C.; Sun, K.; Zhang, C., Quantum phases of Bose-Einstein condensates with synthetic spin-orbital-angular-momentum coupling, Phys. Rev. A, 91, 5, (2015)
[37] Seydaoğlu, M.; Blanes, S., High-order splitting methods for separable non-autonomous parabolic equations, Appl. Numer. Math., 84, 22-32, (2014) · Zbl 1293.65125
[38] Shampine, L. F., Error estimation and control for odes, J. Sci. Comput., 25, 1, 3-16, (2005) · Zbl 1203.65122
[39] Wen, L.; Li, J., Structure and dynamics of a rotating superfluid Bose-Fermi mixture, Phys. Rev. A, 90, 5, (2014)
[40] Yoshida, H., Construction of higher order symplectic integrators, Phys. Lett. A, 150, 5-7, 262-268, (1990)
[41] Zhai, H., Degenerate quantum gases with spin-orbit coupling: a review, Rep. Prog. Phys., 78, 2, (2015)
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