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October  2016, 9(5): 1327-1349. doi: 10.3934/dcdss.2016053

Solving highly-oscillatory NLS with SAM: Numerical efficiency and long-time behavior

Philippe Chartier 1, , Norbert J. Mauser 2, , Florian Méhats 3, and  Yong Zhang 3,

1. 

INRIA-Rennes, IRMAR and ENS Bruz, Campus de Beaulieu, 35042 Rennes Cedex, France
2. 

Wolfgang Pauli Institute c/o Fak. Mathematik, University Wien, Nordbergstrasse 15, 1090 Vienna, Austria
3. 

IRMAR, Université de Rennes 1 and INRIA-Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France, France

Received  October 2014 Revised  July 2015 Published  October 2016

In this paper, we present the Stroboscopic Averaging Method (SAM), recently introduced in [7,8,10,13], which aims at numerically solving highly-oscillatory differential equations. More specifically, we first apply SAM to the Schrödinger equation on the $1$-dimensional torus and on the real line with harmonic potential, with the aim of assessing its efficiency: as compared to the well-established standard splitting schemes, the stiffer the problem is, the larger the speed-up grows (up to a factor $100$ in our tests). The geometric properties of SAM are also explored: on very long time intervals, symmetric implementations of the method show a very good preservation of the mass invariant and of the energy. In a second series of experiments on $2$-dimensional equations, we demonstrate the ability of SAM to capture qualitatively the long-time evolution of the solution (without spurring high oscillations).
Keywords: invariants, Hamiltonian PDEs, nonlinear Schrödinger., stroboscopic averaging, Highly-oscillatory evolution equation.
Mathematics Subject Classification: 34K33, 37L05, 35Q5.
Citation: Philippe Chartier, Norbert J. Mauser, Florian Méhats, Yong Zhang. Solving highly-oscillatory NLS with SAM: Numerical efficiency and long-time behavior. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1327-1349. doi: 10.3934/dcdss.2016053
References:
[1]

W. Bao, S. Jin and P. A. Markowich, On Time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), 487-524. doi: 10.1006/jcph.2001.6956.  Google Scholar

[2]

W. Bao, S. Jin and P. A. Markowich, Numerical studies of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regime, SIAM J. Sci. Comput., 25 (2003), 27-64. doi: 10.1137/S1064827501393253.  Google Scholar

[3]

W. Bao and J. Shen, A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose-Einstein condensates, SIAM J. Sci. Comput., 26 (2005), 2010-2028. doi: 10.1137/030601211.  Google Scholar

[4]

N. Ben Abdallah, Y. Y. Cai, F. Castella and F. Méhats, Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential, Kinetic and Related Models, 4 (2011), 831-856. doi: 10.3934/krm.2011.4.831.  Google Scholar

[5]

N. Ben Abdallah, F. Castella and F. Méhats, Time averaging for the strongly confined nonlinear Schrödinger equation, using almost-periodicity, J. Differ. Equations, 245 (2008), 154-200. doi: 10.1016/j.jde.2008.02.002.  Google Scholar

[6]

S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus and expansion and some of its applications, Physics Reports, 470 (2009), 151-238. doi: 10.1016/j.physrep.2008.11.001.  Google Scholar

[7]

M. P. Calvo, P. Chartier, A. Murua and J. M. Sanz-Serna, A stroboscopic numerical method for highly oscillatory problems, in Numerical Analysis and Multiscale Computations, Lect. Notes Comput. Sci. Eng., 82 (2012), 71-85. doi: 10.1007/978-3-642-21943-6_4.  Google Scholar

[8]

M. P. Calvo, P. Chartier, A. Murua and J.-M. Sanz-Serna, Numerical stroboscopic averaging for ODEs and DAEs, Appl. Numer. Math., 61 (2011), 1077-1095. doi: 10.1016/j.apnum.2011.06.007.  Google Scholar

[9]

R. Carles and E. Faou, Energy cascades for NLS on the torus, Discrete and Continuous Dynamical Systems-Series A (DCDS-A), 32 (2012), 2063-2077. doi: 10.3934/dcds.2012.32.2063.  Google Scholar

[10]

F. Castella, P. Chartier, F. Méhats and A. Murua, Stroboscopic averaging for the nonlinear Schrödinger equations, Foundations of Computational Mathematics, 15 (2015), 519-559. doi: 10.1007/s10208-014-9235-7.  Google Scholar

[11]

P. Chartier, F. Méhats, M. Thalhammer and Y. Zhang, A note on the convergence of splitting methods for periodic highly-oscillatory systems,, preprint., ().   Google Scholar

[12]

P. Chartier, A. Murua and J. M. Sanz-Serna, A formal series approach to averaging: Exponentially small error estimates, Discrete and Continuous Dynamical Systems-Series A (DCDS-A), 32 (2012), 3009-3027. doi: 10.3934/dcds.2012.32.3009.  Google Scholar

[13]

P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration I: B-series, Found Comput Math, 10 (2010), 695-727. doi: 10.1007/s10208-010-9074-0.  Google Scholar

[14]

W. E, Analysis of the heterogeneous multiscale method for ordinary differential equations, Comm. Math. Sci., 1 (2003), 423-436. doi: 10.4310/CMS.2003.v1.n3.a3.  Google Scholar

[15]

W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), 87-132. doi: 10.4310/CMS.2003.v1.n1.a8.  Google Scholar

[16]

W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden, Heterogeneous multiscale methods: A review, Commun. Comput. Phys., 2 (2007), 367-450.  Google Scholar

[17]

B. Engquist and R. Tsai, Heterogeneous multiscale methods for stiff ordinary differential equations, Math. Comput., 74 (2005), 1707-1742. doi: 10.1090/S0025-5718-05-01745-X.  Google Scholar

[18]

B. Grébert and C. Villegas-Blas, On the energy exchange between resonant modes in nonlinear Schrödinger equations, Ann. I. H. Poincaré, 28 (2011), 127-134. doi: 10.1016/j.anihpc.2010.11.004.  Google Scholar

[19]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, 2006.  Google Scholar

[20]

A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48 (1984), 133-139. doi: 10.1016/0021-8928(84)90078-9.  Google Scholar

[21]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Springer, 2007.  Google Scholar

[22]

J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, 2011. doi: 10.1007/978-3-540-71041-7.  Google Scholar

[23]

G. Strang, On the construction and comparision of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517. doi: 10.1137/0705041.  Google Scholar

[24]

M. Thalhammer, M. Caliari and C. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228 (2009), 822-832. doi: 10.1016/j.jcp.2008.10.008.  Google Scholar

[25]

H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268. doi: 10.1016/0375-9601(90)90092-3.  Google Scholar

[26]

Y. Zhang and X. Dong, On the computation of ground state and dynamics of Schrödinger-Poisson-Slater system, J. Comput. Phys., 230 (2011), 2660-2676. doi: 10.1016/j.jcp.2010.12.045.  Google Scholar

show all references

References:
[1]

W. Bao, S. Jin and P. A. Markowich, On Time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), 487-524. doi: 10.1006/jcph.2001.6956.  Google Scholar

[2]

W. Bao, S. Jin and P. A. Markowich, Numerical studies of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regime, SIAM J. Sci. Comput., 25 (2003), 27-64. doi: 10.1137/S1064827501393253.  Google Scholar

[3]

W. Bao and J. Shen, A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose-Einstein condensates, SIAM J. Sci. Comput., 26 (2005), 2010-2028. doi: 10.1137/030601211.  Google Scholar

[4]

N. Ben Abdallah, Y. Y. Cai, F. Castella and F. Méhats, Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential, Kinetic and Related Models, 4 (2011), 831-856. doi: 10.3934/krm.2011.4.831.  Google Scholar

[5]

N. Ben Abdallah, F. Castella and F. Méhats, Time averaging for the strongly confined nonlinear Schrödinger equation, using almost-periodicity, J. Differ. Equations, 245 (2008), 154-200. doi: 10.1016/j.jde.2008.02.002.  Google Scholar

[6]

S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus and expansion and some of its applications, Physics Reports, 470 (2009), 151-238. doi: 10.1016/j.physrep.2008.11.001.  Google Scholar

[7]

M. P. Calvo, P. Chartier, A. Murua and J. M. Sanz-Serna, A stroboscopic numerical method for highly oscillatory problems, in Numerical Analysis and Multiscale Computations, Lect. Notes Comput. Sci. Eng., 82 (2012), 71-85. doi: 10.1007/978-3-642-21943-6_4.  Google Scholar

[8]

M. P. Calvo, P. Chartier, A. Murua and J.-M. Sanz-Serna, Numerical stroboscopic averaging for ODEs and DAEs, Appl. Numer. Math., 61 (2011), 1077-1095. doi: 10.1016/j.apnum.2011.06.007.  Google Scholar

[9]

R. Carles and E. Faou, Energy cascades for NLS on the torus, Discrete and Continuous Dynamical Systems-Series A (DCDS-A), 32 (2012), 2063-2077. doi: 10.3934/dcds.2012.32.2063.  Google Scholar

[10]

F. Castella, P. Chartier, F. Méhats and A. Murua, Stroboscopic averaging for the nonlinear Schrödinger equations, Foundations of Computational Mathematics, 15 (2015), 519-559. doi: 10.1007/s10208-014-9235-7.  Google Scholar

[11]

P. Chartier, F. Méhats, M. Thalhammer and Y. Zhang, A note on the convergence of splitting methods for periodic highly-oscillatory systems,, preprint., ().   Google Scholar

[12]

P. Chartier, A. Murua and J. M. Sanz-Serna, A formal series approach to averaging: Exponentially small error estimates, Discrete and Continuous Dynamical Systems-Series A (DCDS-A), 32 (2012), 3009-3027. doi: 10.3934/dcds.2012.32.3009.  Google Scholar

[13]

P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration I: B-series, Found Comput Math, 10 (2010), 695-727. doi: 10.1007/s10208-010-9074-0.  Google Scholar

[14]

W. E, Analysis of the heterogeneous multiscale method for ordinary differential equations, Comm. Math. Sci., 1 (2003), 423-436. doi: 10.4310/CMS.2003.v1.n3.a3.  Google Scholar

[15]

W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), 87-132. doi: 10.4310/CMS.2003.v1.n1.a8.  Google Scholar

[16]

W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden, Heterogeneous multiscale methods: A review, Commun. Comput. Phys., 2 (2007), 367-450.  Google Scholar

[17]

B. Engquist and R. Tsai, Heterogeneous multiscale methods for stiff ordinary differential equations, Math. Comput., 74 (2005), 1707-1742. doi: 10.1090/S0025-5718-05-01745-X.  Google Scholar

[18]

B. Grébert and C. Villegas-Blas, On the energy exchange between resonant modes in nonlinear Schrödinger equations, Ann. I. H. Poincaré, 28 (2011), 127-134. doi: 10.1016/j.anihpc.2010.11.004.  Google Scholar

[19]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, 2006.  Google Scholar

[20]

A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48 (1984), 133-139. doi: 10.1016/0021-8928(84)90078-9.  Google Scholar

[21]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Springer, 2007.  Google Scholar

[22]

J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, 2011. doi: 10.1007/978-3-540-71041-7.  Google Scholar

[23]

G. Strang, On the construction and comparision of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517. doi: 10.1137/0705041.  Google Scholar

[24]

M. Thalhammer, M. Caliari and C. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228 (2009), 822-832. doi: 10.1016/j.jcp.2008.10.008.  Google Scholar

[25]

H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268. doi: 10.1016/0375-9601(90)90092-3.  Google Scholar

[26]

Y. Zhang and X. Dong, On the computation of ground state and dynamics of Schrödinger-Poisson-Slater system, J. Comput. Phys., 230 (2011), 2660-2676. doi: 10.1016/j.jcp.2010.12.045.  Google Scholar

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