Advertisement

Energy preserving model order reduction of the nonlinear Schrödinger equation

  • Bülent Karasözen
  • Murat Uzunca
Article
  • 69 Downloads

Abstract

An energy preserving reduced order model is developed for two dimensional nonlinear Schrödinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orthogonal decomposition (POD) Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and mass are shown for the full order model (FOM) and for the ROM which ensures the long term stability of the solutions. Numerical simulations illustrate the preservation of the energy and mass in the reduced order model for the two dimensional NLSE with and without the external potential. The POD-DMD makes a remarkable improvement in computational speed-up over the POD-DEIM. Both methods approximate accurately the FOM, whereas POD-DEIM is more accurate than the POD-DMD.

Keywords

Nonlinear Schrödinger equation Discontinuous Galerkin method Average vector field method Proper orthogonal decomposition Discrete empirical interpolation method Dynamic mode decomposition 

Mathematics Subject Classification (2010)

65P10 65M60 365Q55 37M15 93A15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The authors would like to thank the reviewers for the comments and suggestions that helped to improve the manuscript.

References

  1. 1.
    Afkham, B.M., Hesthaven, J.S.: Structure preserving model reduction of parametric Hamiltonian systems. SIAM J. Sci. Comput. 39(6), A2616–A2644 (2017).  https://doi.org/10.1137/17M1111991 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alla, A., Kutz, J.: Randomized model order reduction ArXiv e-prints (2016)Google Scholar
  3. 3.
    Alla, A., Kutz, J.N.: Nonlinear model order reduction via dynamic mode decomposition. SIAM J. Sci. Comput. 39(5), B778–B796 (2017).  https://doi.org/10.1137/16M1059308 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Antil, H., Heinkenschloss, M., Sorensen Danny, C.: Application of the discrete empirical interpolation method to reduced order modeling of nonlinear and parametric systems. In: Quarteroni, A., Rozza, G. (eds.) Reduced Order Methods for Modeling and Computational Reduction, MS & A - Modeling, Simulation and Applications, vol. 9, pp 101–136. Springer International Publishing (2014),  https://doi.org/10.1007/978-3-319-02090-7_4
  5. 5.
    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).  https://doi.org/10.1016/j.cpc.2013.07.012 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    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).  https://doi.org/10.1016/j.cpc.2014.06.026 CrossRefMATHGoogle Scholar
  7. 7.
    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).  https://doi.org/10.1016/j.cpc.2015.03.012 CrossRefMATHGoogle Scholar
  8. 8.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002).  https://doi.org/10.1137/S0036142901384162 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Astrid, P., Weiland, S., Willcox, K., Backx, T.: Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans. Autom. Control 53(10), 2237–2251 (2008).  https://doi.org/10.1109/TAC.2008.2006102 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bao, W., Cai, Y.: Mathematical theory and numerical methods for Bose–Einstein condensation. Kinetic Relat. Models 6(1), 1–135 (2013).  https://doi.org/10.3934/krm.2013.6.1 MathSciNetMATHGoogle Scholar
  11. 11.
    Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique 339(9), 667–672 (2004).  https://doi.org/10.1016/j.crma.2004.08.006 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Beattie, C., Gugercin, S.: Structure-preserving model reduction for nonlinear port-Hamiltonian systems. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference, pp. 6564–6569.  https://doi.org/10.1109/CDC.2011.6161504 (2011)
  13. 13.
    Bistrian, D.A., Navon, I.M.: Randomized dynamic mode decomposition for nonintrusive reduced order modelling. Int. J. Numer. Methods Eng.  https://doi.org/10.1002/nme.5499 (2017)
  14. 14.
    Bridges, T.J., Reich, S.: Numerical methods for Hamiltonian PDEs. J. Phys. A Math. Gen. 39(19), 5287–5320 (2006).  https://doi.org/10.1088/0305-4470/39/19/S02 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Carlberg, K., Farhat, C., Cortial, J., Amsallem, D.: The {GNAT} method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows. J. Comput. Phys. 242, 623–647 (2013).  https://doi.org/10.1016/j.jcp.2013.02.028 MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Carlberg, K., Tuminaro, R., Boggs, P.: Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics. SIAM J. Sci. Comput. 37(2), B153–B184 (2015).  https://doi.org/10.1137/140959602 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Celledoni, E., Owren, B., Sun, Y.: The minimal stage, energy preserving Runge-Kutta method for polynomial Hamiltonian systems is the averaged vector field method. Math. Comp. 83(288), 1689–1700 (2014).  https://doi.org/10.1090/S0025-5718-2014-02805-6 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Celledoni, E., Grimm, V., McLachlan, R.I., McLaren, D.I., O’Neale, D.J., Owren, B., Quispel, G.R.W.: Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method. J. Comput. Phys. 231, 6770–6789 (2012).  https://doi.org/10.1016/j.jcp.2012.06.022 MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Charnyi, S., Heister, T., Olshanskii, M. A., Rebholz, L.G.: On conservation laws of Navier-Stokes Galerkin discretizations. J. Comput. Phys. 337, 289–308 (2017).  https://doi.org/10.1016/j.jcp.2017.02.039 MathSciNetCrossRefGoogle Scholar
  20. 20.
    Chaturantabut, S., Beattie, C., Gugercin, S.: Structure-preserving model reduction for nonlinear Port-Hamiltonian systems. SIAM J. Sci. Comput. 38(5), B837–B865 (2016).  https://doi.org/10.1137/15M1055085 MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010).  https://doi.org/10.1137/090766498 MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Chen, J.B., Qin, M.Z., Tang, Y.F.: Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation. Comput. Math. Appl. 43(8), 1095–1106 (2002).  https://doi.org/10.1016/S0898-1221(02)80015-3 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Cohen, D., Hairer, E.: Linear energy-preserving integrators for Poisson systems. BIT Numer. Math. 51(1), 91–101 (2011).  https://doi.org/10.1007/s10543-011-0310-z MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Debussche, A., Faou, E.: Modified energy for split-step methods applied to the linear Schrödinger equation. SIAM J. Numer. Anal. 47(5), 3705–3719 (2009).  https://doi.org/10.1137/080744578 MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Drohmann, M., Haasdonk, B., Ohlberger, M.: Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34(2), A937–A969 (2012).  https://doi.org/10.1137/10081157X MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Erichson, N.B., Donovan, C.: Randomized low-rank dynamic mode decomposition for motion detection. Comput. Vis. Image Underst. 146, 40–50 (2016).  https://doi.org/10.1016/j.cviu.2016.02.005 CrossRefGoogle Scholar
  27. 27.
    Everson, R., Sirovich, L.: Karhunen–Loève procedure for gappy data. J. Opt. Soc. Am. A 12(8), 1657–1664 (1995).  https://doi.org/10.1364/JOSAA.12.001657 CrossRefGoogle Scholar
  28. 28.
    Galati, L., Zheng, S.: Nonlinear Schrödinger equations for Bose-Einstein condensates. AIP Conf. Proc. 1562(1), 50–64 (2013).  https://doi.org/10.1063/1.4828682 CrossRefGoogle Scholar
  29. 29.
    Gao, Y., Mei, L.: Implicit–explicit multistep methods for general two-dimensional nonlinear Schrödinger equations. Appl. Numer. Math. 106, 41–60 (2016).  https://doi.org/10.1016/j.apnum.2016.06.003 CrossRefMATHGoogle Scholar
  30. 30.
    Gong, Y., Cai, J., Wang, Y.: Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian {PDEs}. J. Comput. Phys. 279, 80–102 (2014).  https://doi.org/10.1016/j.jcp.2014.09.001 MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Gong, Y., Wang, Q., Wang, Z.: Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems. Comput. Methods Appl. Mech. Eng. 315, 780–798 (2017).  https://doi.org/10.1016/j.cma.2016.11.016 MathSciNetCrossRefGoogle Scholar
  32. 32.
    Gong, Y., Wang, Y.: An energy-preserving wavelet collocation method for general multi-symplectic formulations of Hamiltonian PDEs. Commun. Comput. Phys. 20 (5), 1313–1339 (2016).  https://doi.org/10.4208/cicp.231014.110416a MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics. Springer, Heidelberg (2010)MATHGoogle Scholar
  34. 34.
    Halko, N., Martinsson, P.G., Tropp, J.A.: Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53(2), 217–288 (2011).  https://doi.org/10.1137/090771806 MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Islas, A., Karpeev, D., Schober, C.: Geometric integrators for the nonlinear Schrödinger equation. J. Comput. Phys. 173(1), 116–148 (2001).  https://doi.org/10.1006/jcph.2001.6854 MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Karasözen, B., Şimşek, G.: Energy preserving integration of bi-Hamiltonian partial differential equations. Appl. Math. Lett. 26(12), 1125–1133 (2013).  https://doi.org/10.1016/j.aml.2013.06.005 MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Karasözen, B., Akkoyunlu, C., Uzunca, M.: Model order reduction for nonlinear Schrödinger equation. Appl. Math. Comput. 258, 509–519 (2015).  https://doi.org/10.1016/j.amc.2015.02.001 MathSciNetMATHGoogle Scholar
  38. 38.
    Karasözen, B., Küċükseyhan, T., Uzunca, M.: Structure preserving integration and model order reduction of skew-gradient reaction–diffusion systems. Ann. Oper. Res. 258(1), 79–106 (2017).  https://doi.org/10.1007/s10479-015-2063-6 MathSciNetCrossRefGoogle Scholar
  39. 39.
    Karasözen, B., Uzunca, M., Sarıaydın-Filibelioğlu, A., Yücel, H.: Energy stable discontinuous Galerkin finite element method for the Allen-Cahn equation. Int. J. Comput. Methods 0(0), 1850,013 (0) (2017).  https://doi.org/10.1142/S0219876218500135 Google Scholar
  40. 40.
    Koopman, B.O.: Hamiltonian systems and transformation in Hilbert space. Proc. Natl. Acad. Sci. 17(5), 315–318 (1931)CrossRefMATHGoogle Scholar
  41. 41.
    Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90(1), 117–148 (2001).  https://doi.org/10.1007/s002110100282 MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Kutz, J.N., Brunton, S.L., Brunton, B.W., Proctor, J.L.: Dynamic Mode Decomposition: Data-driven Modeling of Complex Systems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2016)CrossRefMATHGoogle Scholar
  43. 43.
    Lall, S., Krysl, P., Marsden, J. E.: Structure-preserving model reduction for mechanical systems. Phys. D 184(1-4), 304–318 (2003).  https://doi.org/10.1016/S0167-2789(03)00227-6 MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Li, Y. W., Wu, X.: General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs. J. Comput. Phys. 301, 141–166 (2015).  https://doi.org/10.1016/j.jcp.2015.08.023 MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Mahoney, M.W.: Randomized algorithms for matrices and data. Found. Trends Mach. Learn. 3(2), 123–224 (2011).  https://doi.org/10.1561/2200000035 MATHGoogle Scholar
  46. 46.
    Martinsson, P.G.: Randomized methods for matrix computations and analysis of high dimensional data ArXiv e-prints (2016)Google Scholar
  47. 47.
    Mezić, I.: Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45(1), 357–378 (2013).  https://doi.org/10.1146/annurev-fluid-011212-140652 MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Mohebujjaman, M., Rebholz, L.G., Xie, X., Iliescu, T.: Energy balance and mass conservation in reduced order models of fluid flows. J. Comput. Phys. 346 (Supplement C), 262–277 (2017).  https://doi.org/10.1016/j.jcp.2017.06.019 MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Peng, L., Mohseni, K.: Symplectic model reduction of Hamiltonian systems. SIAM J. Sci. Comput. 38(1), A1–A27 (2016).  https://doi.org/10.1137/140978922 MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Pitaevskii, L.P., Stringari, S.: Bose-Einstein Condensation. Clarendon Press, Oxford (2003)MATHGoogle Scholar
  51. 51.
    Quispel, G., McLaren, D.: A new class of energy-preserving numerical integration methods. J. Phys. Math. Theor. 41(4), 045206 (7pp) (2008).  https://doi.org/10.1088/1751-8113/41/4/045206 MathSciNetMATHGoogle Scholar
  52. 52.
    Riviere, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM.  https://doi.org/10.1137/1.9780898717440 (2008)
  53. 53.
    Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D. S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009).  https://doi.org/10.1017/S0022112009992059 MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010).  https://doi.org/10.1017/S0022112010001217 MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Sulem, C., Sulem, P.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Applied Mathematical Sciences. Springer, New York (1999)MATHGoogle Scholar
  56. 56.
    Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: Theory and applications. J. Comput. Dyn. 1(2), 391–421 (2014).  https://doi.org/10.3934/jcd.2014.1.391 MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Uzunca, M., Karasözen, B.: Energy stable model order reduction for the Allen-Cahn equation. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds.) Model Reduction of Parametrized Systems, pp 403–419. Springer International Publishing, Cham (2017),  https://doi.org/10.1007/978-3-319-58786-8_25
  58. 58.
    Vemaganti, K.: Discontinuous Galerkin methods for periodic boundary value problems. Numer. Methods Partial Differ. Equ. 23(3), 587–596 (2007).  https://doi.org/10.1002/num.20191 MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Wang, T., Guo, B., Xu, Q.: Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions. J. Comput. Phys. 243, 382–399 (2013).  https://doi.org/10.1016/j.jcp.2013.03.007 MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Williams, M.O., Schmid, P.J., Kutz, J.N.: Hybrid reduced-order integration with proper orthogonal decomposition and dynamic mode decomposition. Multiscale Model. Simul. 11(2), 522–544 (2013).  https://doi.org/10.1137/120874539 MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for nonlinear Schrödinger equations. J. Comput. Phys. 205(1), 72–97 (2005).  https://doi.org/10.1016/j.jcp.2004.11.001 MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Xu, Y., Zhang, L.: Alternating direction implicit method for solving two-dimensional cubic nonlinear Schrödinger equation. Comput. Phys. Commun. 183(5), 1082–1093 (2012).  https://doi.org/10.1016/j.cpc.2012.01.006 MathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Zimmermann, R., Willcox, K.: An accelerated greedy missing point estimation procedure. SIAM J. Sci. Comput. 38(5), A2827–A285 (2016).  https://doi.org/10.1137/15M1042899 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Applied Mathematics & Department of MathematicsMiddle East Technical UniversityAnkaraTurkey
  2. 2.Department of MathematicsSinop UniversitySinopTurkey

Personalised recommendations