Mechthild Thalhammer- High-order time-splitting spectral methods for Gross–Pitaevskii systems

IntroductionGross–Pitaevskii equation

Spatial discretisationTime integration

Conclusions

High-order time-splitting spectral methodsfor Gross–Pitaevskii systems

Mechthild Thalhammer, University of Innsbruck, Austria

Mathematical Models of Quantum FluidsSeptember 2009, Verona, Italy

Mechthild Thalhammer Splitting spectral methods for GPS



Conclusions

Bose–Einstein condensation

In our laboratories temperatures are measuredin micro- or nanokelvin ... In this ultracoldworld ... atoms move at a snail’s pace ... andbehave like matter waves. Interesting andfascinating new states of quantum matter areformed and investigated in our experiments.

Grimm et al.

Practical realisation. Observation of Bose–Einstein condensationin physical experiments.

Theoretical model. Mathematical description by systems ofnonlinear Schrödinger equations.

Numerical simulation. Favourable discretisations rely ontime-splitting pseudospectral methods. See work by BAO, CANCÈS,DION, DU, JAKSCH, MARKOWICH, PÉREZ-GARCÍA, SHEN, TANG,TSUBOTA, TIWARI, SHUKLA, VAZQUEZ, ZHANG etc.




Conclusions

Discretisation of nonlinear Schrödinger equations

Theoretical model. Mathematical description of Bose–Einsteincondensate by Gross–Pitaevskii equation

iħ ∂tψ(x, t ) =(− ħ2

2m ∆+U (x)+ 4πħ2aNm |ψ(x, t )|2

)ψ(x, t ) .

Numerical discretisation. Highaccuracy approximations rely on

Hermite and Fourier spectralmethods in space and

exponential operator splittingmethods in time. −3

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Conclusions

Objectives

Convergence analysis.

High-order splitting methodsfor nonlinear Schrödingerequations.

Minimisation method forground state computation.

Implementation.

Numerical simulation ofGross–Pitaevskii systems inthree space dimensions(ground state, time evolution).

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Conclusions

Contents

Contents.

Gross–Pitaevskii equation

Pseudospectral methods

Exponential operator splitting methods

Linear evolutionary Schrödinger equationsNonlinear evolutionary Schrödinger equationsStability and convergence analysis




Conclusions





Conclusions


Nonlinear Schrödinger equation. Normalised Gross–Pitaevskiiequation for ψ :Rd × [0,∞) →C

i∂tψ(x, t ) =(− 1

2 ∆+V (x)+ϑ |ψ(x, t )|2)ψ(x, t ) ,

∆=d∑

j=1∂2

x j, V (x) = 1

2 VH (x) = 12

d∑j=1

γ4j x2

j ,

subject to asympotic boundary conditions and initial condition

‖ψ(·,0)‖2L2 = 1.

Geometric properties. Preservation of particle number ‖ψ(·, t )‖2L2

and energy functional

E(ψ(·, t )

)= ((− 12 ∆+V + 1

2 ϑ |ψ(·, t )|2)ψ(·, t )∣∣∣ψ(·, t )

)L2

.




Conclusions

Ground state solution

Nonlinear Schrödinger equation. Gross–Pitaevskii equation

i∂tψ(x, t ) =(− 1

2 ∆+V (x)+ϑ |ψ(x, t )|2)ψ(x, t ) .

Ground state. Solution of form

ψ(x, t ) = e− iµt φ(x)

that minimises energy functional.−15 −10 −5 0 5 10 150

0.2

0.4

0.6

0.8ϑ = 1ϑ = 10ϑ = 100ϑ = 1000

Minimisation approach. Compute ground state by directminimisation of energy functional, see BAO AND TANG (2003)

and CALIARI ET AL. (2008). Use ground state solution asreliable reference solution in time-integration.




Conclusions

Hermite pseudospectral methodFourier pseudospectral method

Spatial discretisation




Conclusions


Hermite pseudospectral method

Spectral decomposition. Hermite functions (Hm)m ≥ 0 formorthonormal basis of L2

(Rd

)and satisfy

12

(− ∆+VH)Hm =λm Hm , λm =

d∑j=1

γ2j

(m j + 1

2

).

Hermite decomposition for ψ(·, t ) ∈ L2(Rd

)ψ(·, t ) =∑

mψm(t )Hm , ψm(t ) = (

ψ(·, t ) |Hm)L2 .

Numerical approximation. Truncation of infinite sum andapplication of Gauss–Hermite quadrature formula

ψM (·, t ) = ∑M

mψm(t )Hm ,

ψm(t ) =∫Rdψ(x, t )Hm(x)dx ≈ ∑

Mk

ωk e |ξk |2ψ(ξk , t )Hm(ξk ) .




Conclusions


Fourier pseudospectral method

Spectral decomposition. LetΩ= [−a, a] with a > 0. Fourier basisfunctions (Fm)m∈Zd form orthonormal basis of L2

(Ωd

)and satisfy

− 12 ∆Fm =λm Fm , λm = π2

2 a2

d∑j=1

m2j .

Fourier decomposition for ψ(·, t ) ∈ L2(Ωd

)ψ(·, t ) =∑

mψm(t )Fm , ψm(t ) = (

ψ(·, t ) |Fm)L2 .

Numerical approximation. Truncation of infinite sum andapplication of trapezoid quadrature formula

ψM (·, t ) = ∑M

mψm(t )Fm ,

ψm(t ) =∫Ωdψ(x, t )Fm(x)dx ≈ω∑

Mk

ψ(ξk , t )Fm(ξk ) .




Conclusions

Linear Schrödinger equationsNonlinear Schrödinger equations

Time integration




Conclusions


Evolutionary Schrödinger equations

Evolutionary Schrödinger equations. Formulate nonlinearSchrödinger equations such as Gross–Pitaevskii equation

i∂tψ(x, t ) =(− 1

2 ∆+V (x)+ϑ |ψ(x, t )|2)ψ(x, t ) ,

as abstract differential equation for u(t ) =ψ(·, t )

u′(t ) = A u(t )+B(u(t )

)u(t ) .

Choose differential operator A and multiplication operator B(u)according to spectral space discretisation

i A =− 12

(∆−VH

), iB(u) =V − 1

2 VH +ϑ |u|2 , (Hermite)

i A =− 12 ∆ , iB(u) =V +ϑ |u|2 . (Fourier)

Abstract formulation convenient for construction and theoreticalanalysis of time integration methods.




Conclusions


Splitting methods for linear equations

Aim. For linear evolutionary Schrödinger equation

u′(t ) = A u(t )+B u(t ) , t ≥ 0, u(0) given,

i A =− 12

(∆−VH

), iB =V − 1

2 VH , (Hermite)

i A =− 12 ∆ , iB =V , (Fourier)

determine numerical approximation un ≈ u(tn) at tn = nh.

Approach. Splitting methods rely on suitable composition of

v ′(t ) = A v(t ) , w ′(t ) = B w(t ) .

Spectral decomposition with respect to basis functions (Bm) andpointwise multiplication yields

v(t ) = et A v(0) =∑m

vm e− i t λm Bm , v(0) =∑m

vm Bm ,(w(t )

)(x) = (

etB w(0))(x) = etB(x) (w(0)

)(x) .




Conclusions


Splitting methods for linear equations (Examples)

Lie–Trotter splitting method yields first-order approximation

un+1 = ehB eh A un ≈ u(tn+1) = eh(A+B) u(tn) .

Second-order Strang splitting method given through

un+1 = e12 hB eh A e

12 hB un , un+1 = e

12 h A ehB e

12 h A un .

Higher-order splitting methods by BLANES AND MOAN, KAHAN

AND LI, MCLACHLAN, SUZUKI, and YOSHIDA are cast into form

un+1 =s∏

j=1eb j hB ea j h A un = ebs hB eas h A · · · eb1hB ea1h A un

with real (possible negative) method coefficients (a j ,b j )sj=1.




Conclusions


Convergence analysis

Situation. Exponential operator splitting methods for linearevolutionary Schrödinger equations

u′(t ) = A u(t )+B u(t ) , t ≥ 0,

u(tn+1) = eh(A+B) u(tn) , n ≥ 0, u(0) given,

un+1 =s∏

j=1eb j hB ea j h A un , n ≥ 0, u0 given.

Objective. Derive stiff order conditions and error estimate forgeneral exponential operator splitting method.

Approach. Extend error analysis by JAHNKE AND LUBICH (2000) forsecond-order scheme to splitting methods of arbitrary order.




Conclusions


Convergence result

Theorem (TH. 2007, NEUHAUSER AND TH. 2008)

Suppose that the coefficients of the splitting method fulfill theclassical order conditions for p ≥ 1. Then, provided that the exactsolution is sufficiently regular, the following error estimate holds

‖un −u(tn)‖X ≤C ‖u(0)−u0‖X +C hp , 0 ≤ nh ≤ T .

10−2

10−1

100

10−6

10−4

10−2

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StrangMcLachlan

10−2

100

10−10

10−5

100

YSMBM4−1BM4−2

10−2

10−1

100

10−10

10−5

100

YKLSBM6−1BM6−2BM6−3

Temporal convergence orders of various time-splitting methods for atwo-dimensional linear Schrödinger equation. Error versus stepize.




Conclusions


Sketch of the proof

Approach. Relate global and local error (Lady Windermere’s Fan)

un −u(tn) = Sn (u0 −u(0)

)−n−1∑j=0

Sn− j−1 d j+1,

S =s∏

j=1eb j hB ea j h A , d j+1 = u(t j+1)−S u(t j ).

Deduce stability bound for powers of splitting operator and suitableestimate for local error.

Variation-of-constants formula

Stepwise expansion of etB

Quadrature formulas formultiple integrals

Bounds for iteratedcommutators

Characterise domains ofunbounded operators




Conclusions


Splitting methods for nonlinear equations

Abstract formulation. Rewrite nonlinear Schrödinger equation

i∂tψ(x, t ) =(− 1

2 ∆+V (x)+ϑ |ψ(x, t )|2)ψ(x, t )

as abstract differential equation for u(t ) =ψ(·, t )

u′(t ) = A u(t )+B(u(t )

)u(t ) .


v ′(t ) = A v(t ) , w ′(t ) = B(w(t )

)w(t ) ,

with A,B chosen according to spectral space discretisation

i A =− 12

(∆−VH

), iB(u) =V − 1






Conclusions


Splitting methods for nonlinear equations


v ′(t ) = A v(t ) , w ′(t ) = B(w(t )

)w(t ) ,

i A =− 12

(∆−VH

), iB(u) =V − 1



Spectral decomposition with respect to basis functions (Bm) andinvariance property B

(w(t )

)= B(w(0)

)yields

v(t ) = et A v(0) =∑m

vm e− i t λm Bm , v(0) =∑m

vm Bm ,(w(t )

)(x) = (

etB(w(0)) w(0))(x) = etB(w(0))(x) (w(0)

)(x) .

Theoretical analysis. Employ formal calculus of Lie-derivatives.




Conclusions


Error behaviour (Numerical example)

10−3

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erro

r

StrangMcLachlan

10−3

10−2

10−1

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10−10

10−5

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step size

erro

r

YSMBM4−1BM4−2

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r


10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r

StrangMcLachlan

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r

YSMBM4−1BM4−2

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r


Temporal convergence orders of various time-splitting Hermite (first row) andFourier (second row) pseudospectral methods (GPE in 2d, ϑ= 1, M = 128).




Conclusions


Convergence analysis

Conjecture

Suppose that the coefficients of the splitting method fulfill theclassical order conditions for p ≥ 1. Then, provided that the exactsolution is sufficiently regular, the following error estimate holds

‖un −u(tn)‖X ≤C ‖u(0)−u0‖X +C hp , 0 ≤ nh ≤ T .

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r

StrangMcLachlan

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r

YSMBM4−1BM4−2

10−3

10−2

10−1

100

10−10

10−5

100

step size

erro

r





Conclusions

Conclusions

Contents. High accuracy discretisations ofnonlinear Schrödinger equations bytime-splitting spectral methods.

Convergence analysis for linearevolutionary Schrödinger equations.

Numerical illustrations.

Current research.

Extend error analysis to nonlinearproblems (GPS, MCTDHF equations).

Employ different approach to studytime-splitting methods for linear andnonlinear Schrödinger equations insemi-classical regime.

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Mechthild Thalhammer- High-order time-splitting spectral methods for Gross–Pitaevskii systems

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