The convergence behaviour of multi-revolution composition methods combined with time-splitting methods is analysed for highly oscillatory linear differential equations of Schrödinger type. Numerical experiments illustrate and complement the theoretical investigations.
Accepté le :
DOI : 10.1051/m2an/2017010
Mots-clés : Highly oscillatory differential equations, time-dependent Schrödinger equations, multi-revolution composition methods, operator splitting methods, local error, convergence
@article{M2AN_2017__51_5_1859_0,
author = {Chartier, Philippe and M\'ehats, Florian and Thalhammer, Mechthild and Zhang, Yong},
title = {Convergence of multi-revolution composition time-splitting methods for highly oscillatory differential equations of {Schr\"odinger} type},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1859--1882},
publisher = {EDP-Sciences},
volume = {51},
number = {5},
year = {2017},
doi = {10.1051/m2an/2017010},
zbl = {1421.65011},
mrnumber = {3731552},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2017010/}
}
TY - JOUR AU - Chartier, Philippe AU - Méhats, Florian AU - Thalhammer, Mechthild AU - Zhang, Yong TI - Convergence of multi-revolution composition time-splitting methods for highly oscillatory differential equations of Schrödinger type JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1859 EP - 1882 VL - 51 IS - 5 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2017010/ DO - 10.1051/m2an/2017010 LA - en ID - M2AN_2017__51_5_1859_0 ER -
%0 Journal Article %A Chartier, Philippe %A Méhats, Florian %A Thalhammer, Mechthild %A Zhang, Yong %T Convergence of multi-revolution composition time-splitting methods for highly oscillatory differential equations of Schrödinger type %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1859-1882 %V 51 %N 5 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2017010/ %R 10.1051/m2an/2017010 %G en %F M2AN_2017__51_5_1859_0
Chartier, Philippe; Méhats, Florian; Thalhammer, Mechthild; Zhang, Yong. Convergence of multi-revolution composition time-splitting methods for highly oscillatory differential equations of Schrödinger type. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1859-1882. doi : 10.1051/m2an/2017010. http://archive.numdam.org/articles/10.1051/m2an/2017010/
R.A. Adams, Sobolev Spaces. Academic Press, Orlando, Florida (1975). | MR | Zbl
W. Bao, Mathematical models and numerical methods for Bose–Einstein condensation. In Vol. IV of Proc. Inter. Congress Math. Seoul (2014) 971–996. | MR
, , and , On the Gross–Pitaevski equation with strongly anisotropic confinement: formal asymptotics and numerical experiments. M3AS 15 (2005) 767–782. | MR | Zbl
, , and , The nonlinear Schrödinger equation with strong anisotropic harmonic potential. SIAM J. Math. Anal. 37 (2005) 189–199. | DOI | MR | Zbl
and , Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods. J. Comput. Appl. Math. 142 (2002) 313–330. | DOI | MR | Zbl
and , Energy cascade for NLS on the torus. DCDS-A 32 (2012) 2063–2077. | DOI | MR | Zbl
, , and , Multi-revolution composition methods for highly oscillatory differential equations. Numer. Math. 128 (2014) 167–192. | DOI | MR | Zbl
, , and , Improved error estimates for splitting methods applied to highly-oscillatory nonlinear Schrödinger equations. Math. Comput. 85 (2016) 2863–2885. | DOI | MR | Zbl
, , and , Solving highly-oscillatory NLS with SAM: numerical efficiency and geometric properties. Discrete Contin. Dyn. Systems – Ser. S 9 (2016) 1327–1349. | DOI | MR | Zbl
and , Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential. Numer. Math. 108 (2007) 223–262. | DOI | MR | Zbl
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000). | MR | Zbl
, and , Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation. Forum Math. Sigma 2 (2014). Available at: | DOI | MR | Zbl
, Convergence of a split-step Hermite method for the Gross–Pitaevskii equation. IMA J. Numer. Anal. 31 (2011) 396–415. | DOI | MR | Zbl
and , Resonant dynamics for the quintic nonlinear Schrödinger equation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 29 (2012) 455–477. | DOI | Numdam | MR | Zbl
and , On the energy exchange between resonant modes in nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 28 (2011) 127–134. | DOI | Numdam | MR | Zbl
, and , Convergence analysis of high-order time-splitting pseudo-spectral methods for Gross–Pitaevskii equations with rotation term. Numer. Math. 127 (2014) 315–364. | DOI | MR | Zbl
, On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77 (2008) 2141–2153. | DOI | MR | Zbl
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Appl. Math. Sci. Springer, New York 44 (1983). | MR | Zbl
, Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schrödinger equations. SIAM J. Numer. Anal. 50 (2013) 3231–3258. | DOI | MR | Zbl
H. Triebel, Higher Analysis. Barth, Leipzig–Berlin–Heidelberg (1992). | MR | Zbl
, Weak second order multi-revolution composition methods for highly oscillatory stochastic differential equations with additive or multiplicative noise. SIAM J. Sci. Comput. 36 (2014) 1770–1796. | DOI | MR | Zbl
, Construction of higher order symplectic integrators. Phys. Lett. A 150 (1990) 262–268. | DOI | MR
Cité par Sources :
