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Energy preserving model order reduction of the nonlinear Schrödinger equation. (English) Zbl 1404.65310
Summary: 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.

MSC:
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
93A15 Large-scale systems
Software:
GPELab
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[1] Afkham, BM; Hesthaven, JS, Structure preserving model reduction of parametric Hamiltonian systems, SIAM J. Sci. Comput., 39, a2616-a2644, (2017) · Zbl 1379.78019
[2] Alla, A., Kutz, J.: Randomized model order reduction ArXiv e-prints (2016)
[3] Alla, A.; Kutz, JN, Nonlinear model order reduction via dynamic mode decomposition, SIAM J. Sci. Comput., 39, b778-b796, (2017) · Zbl 1373.65090
[4] Antil, Harbir; Heinkenschloss, Matthias; Sorensen, Danny C., Application of the Discrete Empirical Interpolation Method to Reduced Order Modeling of Nonlinear and Parametric Systems, 101-136, (2014), Cham · Zbl 1312.65180
[5] Antoine, X.; Bao, W.; Besse, C., Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., 184, 2621-2633, (2013) · Zbl 1344.35130
[6] Antoine, X.; Duboscq, R., GPELab, a Matlab toolbox to solve Gross-Pitaevskii equations i: Computation of stationary solutions, Comput. Phys. Commun., 185, 2969-2991, (2014) · Zbl 1348.35003
[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) · Zbl 1344.82004
[8] Arnold, DN; Brezzi, F.; Cockburn, B.; Marini, LD, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 1749-1779, (2002) · Zbl 1008.65080
[9] Astrid, P.; Weiland, S.; Willcox, K.; Backx, T., Missing point estimation in models described by proper orthogonal decomposition, IEEE Trans. Autom. Control, 53, 2237-2251, (2008) · Zbl 1367.93110
[10] Bao, W.; Cai, Y., Mathematical theory and numerical methods for Bose-Einstein condensation, Kinetic Relat. Models, 6, 1-135, (2013) · Zbl 1266.82009
[11] Barrault, M.; Maday, Y.; Nguyen, NC; Patera, AT, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique, 339, 667-672, (2004) · Zbl 1061.65118
[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] Bistrian, Diana Alina; Navon, Ionel Michael, Randomized dynamic mode decomposition for nonintrusive reduced order modelling, International Journal for Numerical Methods in Engineering, 112, 3-25, (2017)
[14] Bridges, TJ; Reich, S., Numerical methods for Hamiltonian PDEs, J. Phys. A Math. Gen., 39, 5287-5320, (2006) · Zbl 1090.65138
[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) · Zbl 1299.76180
[16] Carlberg, K.; Tuminaro, R.; Boggs, P., Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics, SIAM J. Sci. Comput., 37, b153-b184, (2015) · Zbl 1320.65193
[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, 1689-1700, (2014) · Zbl 1296.65182
[18] Celledoni, E.; Grimm, V.; McLachlan, RI; McLaren, DI; O’Neale, DJ; Owren, B.; Quispel, GRW, Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method, J. Comput. Phys., 231, 6770-6789, (2012) · Zbl 1284.65184
[19] Charnyi, S.; Heister, T.; Olshanskii, MA; Rebholz, LG, On conservation laws of Navier-Stokes Galerkin discretizations, J. Comput. Phys., 337, 289-308, (2017)
[20] Chaturantabut, S.; Beattie, C.; Gugercin, S., Structure-preserving model reduction for nonlinear Port-Hamiltonian systems, SIAM J. Sci. Comput., 38, b837-b865, (2016) · Zbl 1376.37120
[21] Chaturantabut, S.; Sorensen, DC, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32, 2737-2764, (2010) · Zbl 1217.65169
[22] Chen, JB; Qin, MZ; Tang, YF, Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation, Comput. Math. Appl., 43, 1095-1106, (2002) · Zbl 1050.65127
[23] Cohen, D.; Hairer, E., Linear energy-preserving integrators for Poisson systems, BIT Numer. Math., 51, 91-101, (2011) · Zbl 1216.65175
[24] Debussche, A.; Faou, E., Modified energy for split-step methods applied to the linear Schrödinger equation, SIAM J. Numer. Anal., 47, 3705-3719, (2009) · Zbl 1209.65137
[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, a937-a969, (2012) · Zbl 1259.65133
[26] Erichson, NB; Donovan, C., Randomized low-rank dynamic mode decomposition for motion detection, Comput. Vis. Image Underst., 146, 40-50, (2016)
[27] Everson, R.; Sirovich, L., Karhunen-Loève procedure for gappy data, J. Opt. Soc. Am. A, 12, 1657-1664, (1995)
[28] Galati, L.; Zheng, S., Nonlinear Schrödinger equations for Bose-Einstein condensates, AIP Conf. Proc., 1562, 50-64, (2013)
[29] Gao, Y.; Mei, L., Implicit-explicit multistep methods for general two-dimensional nonlinear Schrödinger equations, Appl. Numer. Math., 106, 41-60, (2016) · Zbl 1348.65142
[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) · Zbl 1352.65647
[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)
[32] Gong, Y.; Wang, Y., An energy-preserving wavelet collocation method for general multi-symplectic formulations of Hamiltonian PDEs, Commun. Comput. Phys., 20, 1313-1339, (2016) · Zbl 1388.65120
[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) · Zbl 1228.65237
[34] Halko, N.; Martinsson, PG; Tropp, JA, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53, 217-288, (2011) · Zbl 1269.65043
[35] Islas, A.; Karpeev, D.; Schober, C., Geometric integrators for the nonlinear Schrödinger equation, J. Comput. Phys., 173, 116-148, (2001) · Zbl 0989.65102
[36] Karasözen, B.; Şimşek, G., Energy preserving integration of bi-Hamiltonian partial differential equations, Appl. Math. Lett., 26, 1125-1133, (2013) · Zbl 1308.35249
[37] Karasözen, B.; Akkoyunlu, C.; Uzunca, M., Model order reduction for nonlinear Schrödinger equation, Appl. Math. Comput., 258, 509-519, (2015) · Zbl 1339.65123
[38] Karasözen, B.; Küċükseyhan, T.; Uzunca, M., Structure preserving integration and model order reduction of skew-gradient reaction-diffusion systems, Ann. Oper. Res., 258, 79-106, (2017) · Zbl 1387.35353
[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, 1850,013 (0), (2017) · Zbl 1404.65173
[40] Koopman, BO, Hamiltonian systems and transformation in Hilbert space, Proc. Natl. Acad. Sci., 17, 315-318, (1931) · JFM 57.1010.02
[41] Kunisch, K.; Volkwein, S., Galerkin proper orthogonal decomposition methods for parabolic problems, Numer. Math., 90, 117-148, (2001) · Zbl 1005.65112
[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)
[43] Lall, S.; Krysl, P.; Marsden, JE, Structure-preserving model reduction for mechanical systems, Phys. D, 184, 304-318, (2003) · Zbl 1041.70011
[44] Li, YW; Wu, X., General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs, J. Comput. Phys., 301, 141-166, (2015) · Zbl 1349.65518
[45] Mahoney, MW, Randomized algorithms for matrices and data, Found. Trends Mach. Learn., 3, 123-224, (2011) · Zbl 1232.68173
[46] Martinsson, P.G.: Randomized methods for matrix computations and analysis of high dimensional data ArXiv e-prints (2016)
[47] Mezić, I., Analysis of fluid flows via spectral properties of the Koopman operator, Annu. Rev. Fluid Mech., 45, 357-378, (2013) · Zbl 1359.76271
[48] Mohebujjaman, M.; Rebholz, LG; Xie, X.; Iliescu, T., Energy balance and mass conservation in reduced order models of fluid flows, J. Comput. Phys., 346, 262-277, (2017) · Zbl 1378.76050
[49] Peng, L.; Mohseni, K., Symplectic model reduction of Hamiltonian systems, SIAM J. Sci. Comput., 38, a1-a27, (2016) · Zbl 1330.65193
[50] Pitaevskii, L.P., Stringari, S.: Bose-Einstein Condensation. Clarendon Press, Oxford (2003) · Zbl 1110.82002
[51] Quispel, G.; McLaren, D., A new class of energy-preserving numerical integration methods, J. Phys. Math. Theor., 41, 045206 (7pp), (2008) · Zbl 1132.65065
[52] Riviere, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM. https://doi.org/10.1137/1.9780898717440 (2008) · Zbl 1153.65112
[53] Rowley, CW; Mezić, I.; Bagheri, S.; Schlatter, P.; Henningson, DS, Spectral analysis of nonlinear flows, J. Fluid Mech., 641, 115-127, (2009) · Zbl 1183.76833
[54] Schmid, PJ, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656, 5-28, (2010) · Zbl 1197.76091
[55] Sulem, C., Sulem, P.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Applied Mathematical Sciences. Springer, New York (1999) · Zbl 0928.35157
[56] Tu, JH; Rowley, CW; Luchtenburg, DM; Brunton, SL; Kutz, JN, On dynamic mode decomposition: Theory and applications, J. Comput. Dyn., 1, 391-421, (2014) · Zbl 1346.37064
[57] Uzunca, Murat; Karasözen, Bülent, Energy Stable Model Order Reduction for the Allen-Cahn Equation, 403-419, (2017), Cham · Zbl 06861112
[58] Vemaganti, K., Discontinuous Galerkin methods for periodic boundary value problems, Numer. Methods Partial Differ. Equ., 23, 587-596, (2007) · Zbl 1114.65144
[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) · Zbl 1349.65347
[60] Williams, MO; Schmid, PJ; Kutz, JN, Hybrid reduced-order integration with proper orthogonal decomposition and dynamic mode decomposition, Multiscale Model. Simul., 11, 522-544, (2013) · Zbl 1293.65136
[61] Xu, Y.; Shu, CW, Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys., 205, 72-97, (2005) · Zbl 1072.65130
[62] Xu, Y.; Zhang, L., Alternating direction implicit method for solving two-dimensional cubic nonlinear Schrödinger equation, Comput. Phys. Commun., 183, 1082-1093, (2012) · Zbl 1277.65073
[63] Zimmermann, R.; Willcox, K., An accelerated greedy missing point estimation procedure, SIAM J. Sci. Comput., 38, a2827-a285, (2016) · Zbl 1348.65045
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