Exponential time integration of solitary waves of cubic Schrödinger equation

The aim of the present paper is to study the suitability of using exponential methods for the time integration of cubic Schrodinger equation till long times. We center on second-order methods, for which we prove a higher order of accuracy on the main invariants when integrating solitary waves. Some geometric implicit exponential methods are considered as well as some explicit suitably projected ones. The comparison in terms of efficiency is performed.

[1]  Brynjulf Owren,et al.  Solving the nonlinear Schrodinger equation using exponential integrators , 2006 .

[2]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[3]  Paolo Novati,et al.  Using the Restricted-denominator Rational Arnoldi Method for Exponential Integrators , 2011, SIAM J. Matrix Anal. Appl..

[4]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[5]  W. Jason,et al.  アルゴリズム869:ODRPACK95:範囲制約のある重み付け直交距離回帰コード , 2007 .

[6]  B. Cano,et al.  High-order symmetric multistep cosine methods , 2013 .

[7]  Manuel Calvo,et al.  Approximate preservation of quadratic first integrals by explicit Runge–Kutta methods , 2010, Adv. Comput. Math..

[8]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[9]  B. Cano,et al.  Multistep cosine methods for second-order partial differential systems , 2010 .

[10]  Brynjulf Owren,et al.  B-series and Order Conditions for Exponential Integrators , 2005, SIAM J. Numer. Anal..

[11]  B. Cano,et al.  Projected explicit lawson methods for the integration of Schrödinger equation , 2015 .

[12]  J. D. Lawson.  Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants , 1967 .

[13]  Philippe Chartier,et al.  Pseudo-symplectic Runge-Kutta methods , 1998 .

[14]  A. Durán,et al.  The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation , 2000 .

[15]  J. M. Sanz-Serna,et al.  Stability and convergence at the PDE/stiff ODE interface , 1989 .

[16]  W. Gragg,et al.  Repeated extrapolation to the limit in the numerical solution of ordinary differential equations , 1964 .

[17]  Håvard Berland,et al.  NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET , 2005 .

[18]  B. Cano,et al.  Conserved quantities of some Hamiltonian wave equations after full discretization , 2006, Numerische Mathematik.

[19]  B. Cano,et al.  Error growth in the numerical integration of periodic orbits by multistep methods, with application to reversible systems , 1998 .

[20]  Marlis Hochbruck,et al.  Explicit Exponential Runge-Kutta Methods for Semilinear Parabolic Problems , 2005, SIAM J. Numer. Anal..

[21]  Elena Celledoni,et al.  Symmetric Exponential Integrators with an Application to the Cubic Schrödinger Equation , 2008, Found. Comput. Math..

[22]  M. Hochbruck,et al.  Exponential Runge--Kutta methods for parabolic problems , 2005 .

[23]  Ernst Hairer,et al.  Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations , 2008, Numerische Mathematik.