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

Previous Article
The IDSA and the homogeneous sphere: Issues and possible improvements
DCDS-S Home
This Issue
Next Article
A multiscale finite element method for Neumann problems in porous microstructures

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

Abstract
Related Papers

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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	P. Chartier, F. Méhats, M. Thalhammer and Y. Zhang, A note on the convergence of splitting methods for periodic highly-oscillatory systems, preprint.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, 2006.
[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.
[21]	J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Springer, 2007.
[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.
[23]	G. Strang, On the construction and comparision of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517. doi: 10.1137/0705041.
[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.
[25]	H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268. doi: 10.1016/0375-9601(90)90092-3.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	P. Chartier, F. Méhats, M. Thalhammer and Y. Zhang, A note on the convergence of splitting methods for periodic highly-oscillatory systems, preprint.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[19]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, 2006.
[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.
[21]	J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Springer, 2007.
[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.
[23]	G. Strang, On the construction and comparision of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517. doi: 10.1137/0705041.
[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.
[25]	H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268. doi: 10.1016/0375-9601(90)90092-3.
[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.

[1]	Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou, Florian Méhats. Averaging of highly-oscillatory transport equations. Kinetic and Related Models, 2020, 13 (6) : 1107-1133. doi: 10.3934/krm.2020039
[2]	Naoufel Ben Abdallah, Yongyong Cai, Francois Castella, Florian Méhats. Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential. Kinetic and Related Models, 2011, 4 (4) : 831-856. doi: 10.3934/krm.2011.4.831
[3]	Claude Le Bris, Frédéric Legoll. Integrators for highly oscillatory Hamiltonian systems: An homogenization approach. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 347-373. doi: 10.3934/dcdsb.2010.13.347
[4]	Emmanuel Frénod, Sever A. Hirstoaga, Eric Sonnendrücker. An exponential integrator for a highly oscillatory vlasov equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 169-183. doi: 10.3934/dcdss.2015.8.169
[5]	Guan Huang. An averaging theorem for nonlinear Schrödinger equations with small nonlinearities. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3555-3574. doi: 10.3934/dcds.2014.34.3555
[6]	Peng Gao, Yong Li. Averaging principle for the Schrödinger equations^†. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2147-2168. doi: 10.3934/dcdsb.2017089
[7]	D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[8]	Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807
[9]	Tarek Saanouni. Remarks on the damped nonlinear Schrödinger equation. Evolution Equations and Control Theory, 2020, 9 (3) : 721-732. doi: 10.3934/eect.2020030
[10]	Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[11]	Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
[12]	Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[13]	Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022039
[14]	Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377
[15]	Partha Guha, Indranil Mukherjee. Hierarchies and Hamiltonian structures of the Nonlinear Schrödinger family using geometric and spectral techniques. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1677-1695. doi: 10.3934/dcdsb.2018287
[16]	Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032
[17]	Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104
[18]	Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107
[19]	Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[20]	Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639

2021 Impact Factor: 1.865

Tools

Metrics

Other articles
by authors

on AIMS
Philippe Chartier Norbert J. Mauser Florian Méhats Yong Zhang
on Google Scholar
Philippe Chartier Norbert J. Mauser Florian Méhats Yong Zhang

[Back to Top]