High order three part split symplectic integration schemes

Transcription

1 High order three part spit sympectic integration schemes Haris Skokos Physics Department, Aristote University of Thessaoniki Thessaoniki, Greece E-mai: URL: Work in coaboration with Joshua Bodyfet, Siegfried Egg, Enrico Gerach, Georgios Papamikos This research has been co-financed by the European Union (European Socia Fund ESF) and Greek nationa funds through the Operationa Program "Education and Lifeong Learning" of the Nationa Strategic Reference Framework (NSRF) - Research Funding Program: Thaes. Investing in knowedge society through the European Socia Fund.

2 Sympectic Integrators Disordered attices Outine The quartic Kein-Gordon (KG) disordered attice The disordered discrete noninear Schrödinger equation (DNLS) Different integration schemes for DNLS Concusions

3 Autonomous Hamitonian systems Consider an N degree of freedom autonomous Hamitonian system having a Hamitonian function of the form: positions momenta H(q 1,q,,q N, p 1,p,,p N ) The time evoution of an orbit (trajectory) with initia condition P(0)=(q 1 (0), q (0),,q N (0), p 1 (0), p (0),,p N (0)) is governed by the Hamiton s equations of motion dpi H dqi H = -, = dt q dt p i i

4 Sympectic Integration schemes Formay the soution of the Hamiton equations of motion can be written as: n dx t n tlh = H, X = LHX X(t) = LHX = e X dt n! where X is the fu coordinate vector and L H the Poisson operator: N H f H f LH f = - j=1 p j q j q j p j If the Hamitonian H can be spit into two integrabe parts as H=A+B, a sympectic scheme for integrating the equations of motion from time t to time t+τ consists of approximating the operator by j n0 e τl H τlh τ(l A +L B ) ciτla diτlb n+1 e = e e e + O(τ ) i=1 for appropriate vaues of constants c i, d i. This is an integrator of order n. So the dynamics over an integration time step τ is described by a series of successive acts of Hamitonians A and B.

5 Sympectic Integrator SABA C e L H The operator can be approximated by the sympectic integrator [Laskar & Robute, Ce. Mech. Dyn. Astr. (001)]: c L d L c L d L c L SABA = e e e e e A 1 B A 1 B 1 A with c 1 = -, c =, d 1 =. 6 3 The integrator has ony sma positive steps and its error is of order. In the case where A is quadratic in the momenta and B depends ony on the positions the method can be improved by introducing a corrector C, having a sma negative step: 3 c - L A,B,B with - 3 c =. 4 C = e Thus the fu integrator scheme becomes: SABAC = C (SABA ) C and its error is of order 4.

6 Interpay of disorder and noninearity Waves in disordered media Anderson ocaization [Anderson, Phys. Rev. (1958)]. Experiments on BEC [Biy et a., Nature (008)] Waves in noninear disordered media ocaization or deocaization? Theoretica and/or numerica studies [Shepeyansky, PRL (1993) Moina, Phys. Rev. B (1998) - Pikovsky & Shepeyansky, PRL (008) - Kopidakis et a., PRL (008) - Fach et a., PRL (009) - Ch.S. et a., PRE (009) - Ch.S. & Fach, PRE (010) Laptyeva et a., EPL (010) - Bodyfet et a., PRE (011) - Bodyfet et a., IJBC (011)] Experiments: propagation of ight in disordered 1d waveguide attices [Lahini et a., PRL, (008)]

7 The Kein Gordon (KG) mode N p ε H = + u + u + u - u 4 W K +1 =1 with fixed boundary conditions u 0 =p 0 =u N+1 =p N+1 =0. Typicay N= Parameters: W and the tota energy E. ε chosen uniformy from,. The discrete noninear Schrödinger (DNLS) equation We aso consider the system: β H = ε ψ - ψ ψ + ψ ψ N 4 * * D + ψ =1 W W where ε chosen uniformy from, and is the noninear parameter. Conserved quantities: The energy and the norm S ψ of the wave packet.

8 Distribution characterization We consider normaized energy distributions in norma mode (NM) space z ν Eν m m of the νth NM. 1 E with E = A + ωa ν ν ν ν N ν ν=1 Second moment:, where A ν is the ampitude m = ν - ν z N with ν = νzν ν=1 Different spreading regimes

9 The KG mode We appy the SABAC integrator scheme to the KG Hamitonian by using the spitting: H K = N = 1 p + ε 1 1 u + u + u - u 4 W A B with a corrector term which corresponds to the Hamitonian function: C= N A,B,B = u ( ε u ) ( u -1 + u+1 - u ). =1 1 W

10 The DNLS mode A nd order SABA Sympectic Integrator with 5 steps, combined with approximate soution for the B part (Fourier Transform): SIFT β 4 1 H = ε ψ + ψ - ψ ψ + ψ ψ ψ = q + ip ε β D q + p + q + p - qnqn+1 -pnpn+ 1 8 H = * * +1, D +1 A B

11 The DNLS mode Sympectic Integrators produced by Successive Spits (SS) H = D ε A β q - q q -p 8 + p + q + p B n n+1 n n+1 p B 1 B Using the SABA integrator we get a nd order integrator with 13 steps, SS : (3-3) (3-3) τl A τ 3τ τ LB LA L τ L 6 B 3 6 SS = e e e e e A ' = / (3-3) (3-3) τ' L B τ' 3τ' τ' τ' L 6 LB L B L 1 B B1 e e e e e (3-3) (3-3) τ' L B τ' 3τ' τ' τ' L 6 LB L B L 1 B B1 e e e e e

12 Non-sympectic methods for the DNLS mode In our study we aso use the DOP853 integrator which is an expicit non-sympectic Runge-Kutta integration scheme of order 8. DOP853: Hairer et a. 1993,

13 Three part spit sympectic integrators for the DNLS mode Three part spit sympectic integrator of order, with 5 steps: ABC H = D ε β q - q q -p 8 + p + q + p n n+1 n n+1 A B C τ τ τ τ L L L L τl A B B A C ABC = e e e e e p This ow order integrator has aready been used by e.g. Chambers, MNRAS (1999) Goździewski et a., MNRAS (008).

14 nd order integrators: Numerica resuts ABC τ=0.005 SS τ=0.0 SIFT τ=0.05 DOP853 δ=10-16 E r : reative energy error S r : reative norm error

15 4 th order sympectic integrators Starting from any nd order sympectic integrator S nd, we can construct a 4 th order integrator S 4th using a composition method [Yoshida, Phys. Let. A (1990)]: S (τ) = S (x τ) S (x τ) S (x τ) 4th nd nd nd / /3 1 1/3 x = -, x = - - Starting with the nd order integrators SS and ABC we construct the 4 th order integrators: SS 4 with 37 steps ABC 4 with 13 steps

16 6 th order sympectic integrators As a higher order integrator, we use the 6 th order sympectic integrator ABC 6 having 9 steps [Yoshida, Phys. Let. A (1990)]: 6 ABC (τ) = ABC (w3τ) ABC (wτ) ABC (w1τ) ABC (w τ) ABC (w τ) ABC (w τ) ABC (w τ) whose coefficients w = w = 1- (w w w ) cannot be given in anaytic form. 1 w = w =

17 High order integrators: Numerica resuts SIFT τ=0.05 SS 4 τ=0.1 ABC 4 τ=0.05 ABC 6 τ=0.15 E r : reative energy error S r : reative norm error

18 Summary We presented severa efficient integration methods suitabe for the integration of the DNLS mode, which are based on sympectic integration techniques. The construction of sympectic schemes based on 3 part spit of the Hamitonian was emphasized (ABC methods). A systematic way of constructing high order ABC integrators was presented. The 4 th and 6 th order integrators proved to be quite efficient, aowing integration of the DNLS for very ong times. We hope that our resuts wi initiate future research both for the theoretica deveopment of new, improved 3 part spit integrators, as we as for their appications to different dynamica systems. Ch.S., Gerach, Bodyfet, Papamikos, Egg (013) arxiv: