-
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
Solving highly-oscillatory NLS with SAM: Numerical efficiency and long-time behavior
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 |
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
[Back to Top]