Mechthild Thalhammer- High-order time-splitting spectral methods for Gross–Pitaevskii systems
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Transcript of 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
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
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.
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
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
−2−1
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x1
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Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
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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−6 −4 −2 0 2 4 6
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Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Contents
Contents.
Gross–Pitaevskii equation
Pseudospectral methods
Exponential operator splitting methods
Linear evolutionary Schrödinger equationsNonlinear evolutionary Schrödinger equationsStability and convergence analysis
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Gross–Pitaevskii equation
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Gross–Pitaevskii equation
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
.
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
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.
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Hermite pseudospectral methodFourier pseudospectral method
Spatial discretisation
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Hermite pseudospectral methodFourier pseudospectral method
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 ) .
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Hermite pseudospectral methodFourier pseudospectral method
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 ) .
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Linear Schrödinger equationsNonlinear Schrödinger equations
Time integration
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Linear Schrödinger equationsNonlinear Schrödinger equations
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.
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Linear Schrödinger equationsNonlinear Schrödinger equations
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) .
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Linear Schrödinger equationsNonlinear Schrödinger equations
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.
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Linear Schrödinger equationsNonlinear Schrödinger equations
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.
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Linear Schrödinger equationsNonlinear Schrödinger equations
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
100
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.
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Linear Schrödinger equationsNonlinear Schrödinger equations
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
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Linear Schrödinger equationsNonlinear Schrödinger equations
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 ) .
Approach. Splitting methods rely on suitable composition of
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
2 VH +ϑ |u|2 , (Hermite)
i A =− 12 ∆ , iB(u) =V +ϑ |u|2 . (Fourier)
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Linear Schrödinger equationsNonlinear Schrödinger equations
Splitting methods for nonlinear equations
Approach. Splitting methods rely on suitable composition of
v ′(t ) = A v(t ) , w ′(t ) = B(w(t )
)w(t ) ,
i A =− 12
(∆−VH
), iB(u) =V − 1
2 VH +ϑ |u|2 , (Hermite)
i A =− 12 ∆ , iB(u) =V +ϑ |u|2 . (Fourier)
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.
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Linear Schrödinger equationsNonlinear Schrödinger equations
Error behaviour (Numerical example)
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
YKLSBM6−1BM6−2BM6−3
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
YKLSBM6−1BM6−2BM6−3
Temporal convergence orders of various time-splitting Hermite (first row) andFourier (second row) pseudospectral methods (GPE in 2d, ϑ= 1, M = 128).
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
Conclusions
Linear Schrödinger equationsNonlinear Schrödinger equations
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
YKLSBM6−1BM6−2BM6−3
Mechthild Thalhammer Splitting spectral methods for GPS
IntroductionGross–Pitaevskii equation
Spatial discretisationTime integration
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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z
Mechthild Thalhammer Splitting spectral methods for GPS