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8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 1/21
Linear Schrödinger equationsNonlinear Schrödinger equations
Convergence of high-order time–splittingpseudospectral methods for Schrödinger equations
Ch. Neuhauser and M. Thalhammer
Department of Mathematics, University of Innsbruck
Three days on Mathematical Models of Quantum fluidsVerona, Italy, 2009
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 2/21
Linear Schrödinger equationsNonlinear Schrödinger equations
Objectives
Gross–Pitaevskii equation (GPE). Nonlinear Schrödingerequation of the form
i ∂ t ψ(x , t ) =− 2
2m∆ + U (x ) + g |ψ(x , t )|2
ψ(x , t ) .
Numerical solution.Space discretisation based on pseudospectral methods.
Time discretisation based on exponential operator splittingmethods.
Aims.Convergence analysis for linear Schrödinger equations.
Comparison of high-order splitting methods regardingaccuracy, efficiency and conservation of geometric properties(particle number, energy).
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 3/21
Linear Schrödinger equationsNonlinear Schrödinger equations
Evolutionary Schrödinger equationsExponential operator splittingConvergence analysis
Linear Schrödinger equations
Time–dependent Schrödinger equations. Normalised linearSchrödinger equation
i ∂ t ψ(x , t ) =−1
2∆ + 1
2 V H(x ) + W (x )
ψ(x , t )
with unbounded polynomial potential
V H(x ) =d
j =1
γ 4 j x 2 j , W (x ) =m∈Nd
αmx m .
Characteristics.
Partial differential equation separable into two parts.
Solution for each part computable in an efficient way.
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 4/21
Linear Schrödinger equationsNonlinear Schrödinger equations
Evolutionary Schrödinger equationsExponential operator splittingConvergence analysis
Abstract evolution equations
Abstract formulation. Interpret time-dependent linearSchrödinger equation as abstract evolution equation
u (t ) =
A + B
u (t ) .
Numerical solution. Split right hand side into two parts
A = i12
∆− V H
, B = −i W .
Numerical method relies on solution of subproblems
v (t ) = A v (t )
v (0) = v 0 given
w (t ) = B w (t )
w (0) = w 0 given
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
E l i S h ödi i
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 5/21
Linear Schrödinger equationsNonlinear Schrödinger equations
Evolutionary Schrödinger equationsExponential operator splittingConvergence analysis
Exponential operator splitting methods
Problem class. Linear evolution equation
u (t ) = A u (t ) + B u (t ), t ≥ 0, u (0) given.
Compute numerical approximation u n ≈ u (t n) at time t n = n h.
Example method. Strang or symmetric Lie–Trotter splitting
u n+1 = eh2AehB
eh2Au n,
see Trotter (1959) and Strang (1968).
Method class. Higher-order exponential operator splitting
methods of the form
u n+1 =s
j =1
ea j hA e
b j hB u n
with coefficients a j , b j (1 ≤ j ≤ s ) .Ch. Neuhauser Convergence of time–splitting pseudospectral methods
E l ti S h ödi ti
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
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Linear Schrödinger equationsNonlinear Schrödinger equations
Evolutionary Schrödinger equationsExponential operator splittingConvergence analysis
Assumptions and hypothesis
Problem class. Linear initial value problem
u (t ) = A u (t ) + B u (t ), t ≥ 0, u (0) given.
Assumptions
Purely imaginary eigenvalues. A : D (A) → L2
(Rd
)generates C 0-group of contraction
etAL2←L2 = 1, ∀t ∈ R.
Real potential. B : D (B ) → L2(Rd ) generates C 0-group of
contraction etB L2←L2 = 1, ∀t ∈ R.
Particle number preservation. A + B : D (A + B ) → L2(Rd )generates C 0-group of contraction
et (A+B )L2←L2 = 1, ∀t ∈ R .
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Evolutionary Schrödinger equations
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 7/21
Linear Schrödinger equationsNonlinear Schrödinger equations
Evolutionary Schrödinger equationsExponential operator splittingConvergence analysis
Assumptions and hypothesis
HypothesisCommutator bounds on weighted sobolev space D p +1 ⊂ L2(Rd )
k j =1
ad
µ j
A (B ) eτ j A
v L2≤ C v D p +1
|µ| = p +1−k , 1 ≤ k ≤ p +1
where ad0A(B ) = B and ad j
A(B ) =
A , ad j −1A (B )
, j ≥ 1.
Reasonable assumptions for evolutionary Schrödinger equationwith polynomial potential B = x m and
D p =
v =µ
v µH µ ∈ L2(Rd ) :µ
µ + m p
mp |v µ|
2 ≤ ∞
.
In particular for Lie splitting
A , B
eτ 1Av L2 ≤ C v D 2 ,
B
eτ 2A
B eτ 1A
v L2 ≤ C v D 2 .
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Evolutionary Schrödinger equations
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 8/21
Linear Schrödinger equationsNonlinear Schrödinger equations
Evolutionary Schrödinger equationsExponential operator splittingConvergence analysis
Convergence result
Situation. Exponential operator splitting for linear evolutionarySchrödinger equation
u (t ) = A u (t ) + B u (t ) , t ≥ 0 , u (0) given ,
u n+1 =s
j =1
ea j hA e
b j hB u n , n ≥ 0 , u 0 given.
Theorem (N., Thalhammer (2009))
Suppose that the coefficients of the exponential operator splitting
method satisfy the classical order conditions for p ≥ 1. Then,provided that u (0) ∈ D p +1, the following error estimate holds
u n − u (t n)L2 ≤ C u 0 − u (0)L2 + hp max
0≤τ ≤t nu (τ )D p +1
.
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Evolutionary Schrödinger equations
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 9/21
Linear Schrödinger equationsNonlinear Schrödinger equations
Evolutionary Schrödinger equationsExponential operator splittingConvergence analysis
Do’s and Don’ts
Main tools.
Variation-of-constants formula for expansion of exact solution
u (t ) = et (A+B )u 0 = e
tAu 0 +
t 0
e(t −τ )AB u (τ )dτ .
Stepwise Taylor expansion of
ehB = I + h B +
1
0τ e(1−τ )hB B 2dτ
Quadrature formulas, Taylor series expansions (commutatorbounds).
Don’ts.
Power series expansions of etA =∞
k =0t k
k ! Ak and
etB = ∞
k =0
t k
k !B k .
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Li S h ödi iEvolutionary Schrödinger equations
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 10/21
Linear Schrödinger equationsNonlinear Schrödinger equations
Evolutionary Schrödinger equationsExponential operator splittingConvergence analysis
Local error estimate for Lie splitting
Expansion of exact solution. Variation-of-constant formula yields
u (h) = ehAu 0 +
h0
e(h−τ 1)A B eτ 1A u 0 dτ 1 + R 2 .
Expansion of numerical solution. Stepwise expansion yields
u 1 =ehA u 0 + h e
hA B u 0 + R̂ 2L2 ,
with R̂ 2L2 + R 2L2 ≤ C h2 max0≤τ ≤h u (τ )D 2 .
Local error. Use rectangular rule for
u (h) − u 1L2 ≤
h
0e(h−τ 1)A B eτ 1A u 0 − e
hA B u 0L2 dτ 1+
C h2 max0≤τ ≤h
u (τ )D 2 .
and obtain finally u (h) − u 1L2 ≤ C h2
max0≤τ ≤h u (τ )D 2 .Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Li S h ödi tiGross–Pitaevskii equation
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 11/21
Linear Schrödinger equationsNonlinear Schrödinger equations
qNumerical approximationLong-term integration of GPE
Gross–Pitaevskii equation
Gross–Pitaevskii equation (GPE). Normalised nonlinearSchrödinger equation
i ∂ t ψ(x , t ) =− 1
2 ∆ + V (x ) + ϑ|ψ(x , t )|2
ψ(x , t )
describes wave function of Bose–Einstein condensate.Harmonic trap. Consider physicallyrelevant case of harmonic potentialV (x ) = 1
2 V H (x ) = 12
d j =1 γ 2 j x 2 j .
Geometric properties. Conserva-tion of particle number ψ(·, t )L2
and energy −3
−2
−1
0
1
2
3
0
2
4
6
8
10
12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x1
t
E
ψ(·, t )
= − 1
2∆ + V + 1
2ϑψ(·, t )
2
ψ(·, t )
ψ(·, t )L2
.
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Linear Schrödinger equationsGross–Pitaevskii equation
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 12/21
Linear Schrödinger equationsNonlinear Schrödinger equations
qNumerical approximationLong-term integration of GPE
Evolutionary Schrödinger equations
Abstract formulation. Interpret time-dependent nonlinearSchrödinger equation as abstract evolution equation
u (t ) =
A + B (u (t ))
u (t ) .
Numerical solution. Split right hand side into two parts
Hermite Fourier
A = i12
∆ − V H
A = i
12
∆B (u ) = −i ϑ|u |2 B (u ) = −i ( 1
2 V H + ϑ|u |2)
Numerical method relies on solutions of evolution equations
v (t ) = A v (t )
v (0) = v 0 given
w (t ) = B (w (t )) w (t )
w (0) = w 0 given
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Linear Schrödinger equationsGross–Pitaevskii equation
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 13/21
Linear Schrödinger equationsNonlinear Schrödinger equations
Numerical approximationLong-term integration of GPE
Pseudospectral methods
Hermite pseudospectral method.
Compute transformation matrices in a preprocessing step.
Hermite transform and inverse Hermite transform in 2D can
be realised by two matrix–matrix multiplications.
Complexity of Hermite transform in 2D is O (M 3).
Fourier pseudospectral method.
Use Fast Fourier Transformation (FFT).Complexity of FFT in 2D is O (log(M ) M 2).
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Linear Schrödinger equationsGross–Pitaevskii equation
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 14/21
Linear Schrödinger equationsNonlinear Schrödinger equations
Numerical approximationLong-term integration of GPE
Numerical experiment (CPU-time)
Computation time of spectral methods.
101
102
10−5
10−4
10−3
10−2
degree of freedom
C P U s
e c o n d s
Hermite 1D
Fourier 1D
101
102
10−5
10−4
10−3
10−2
10−1
degree of freedom
C P U s
e c o n d s
Hermite 2D
Fourier 2D
Figure: Computation time of the Hermite and Fourier spectral methodsin one (left picture) and two (right picture) space dimensions usingM = 2i , 4 ≤ i ≤ 8, basis functions in each space direction.
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Linear Schrödinger equations Gross–Pitaevskii equationN i l i i
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
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Linear Schrödinger equationsNonlinear Schrödinger equations
Numerical approximationLong-term integration of GPE
Numerical experiment (spatial error)
10 25 50 100 250 50010
−16
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
102
degree of freedom
e r r o r
ϑ = 1
ϑ = 10
ϑ = 100
ϑ = 1000
10 25 50 100 250 50010
−16
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
102
degree of freedom
e r r o r
ϑ = 1
ϑ = 10
ϑ = 100
ϑ = 1000
Figure: Spatial error of the Hermite (left picture) and Fourier (rightpicture) spectral method for different values of the coupling constant ϑ.
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Linear Schrödinger equations Gross–Pitaevskii equationN i l i ti
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
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g qNonlinear Schrödinger equations
Numerical approximationLong-term integration of GPE
High-order exponential operator splitting methods
method order #compositionsMcLachlan McLachlan p = 2 s = 3
Strang Strang p = 2 s = 2
BM4-1 Blanes & Moan PRKS6 p = 4 s = 7
BM4-2 Blanes & Moan SRKNb 6
p = 4 s = 7
M4 McLachlan p = 4 s = 6
S4 Suzuki p = 4 s = 6
Y4 Yoshida p = 4 s = 4
BM6-1 Blanes & Moan PRKS10 p = 6 s = 11
BM6-2 Blanes & Moan SRKNb 11
p = 6 s = 12
BM6-3 Blanes & Moan SRKNa14
p = 6 s = 15
KL6 Kahan & Li p = 6 s = 10
S6 Suzuki p = 6 s = 26
Y6 Yoshida p = 6 s = 8
Table: Splitting methods of order p involving s compositions.
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Linear Schrödinger equations Gross–Pitaevskii equationNumerical approximation
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 17/21
g qNonlinear Schrödinger equations
Numerical approximationLong-term integration of GPE
Numerical experiment (long-term integration)
Numerical experiment (Caliari, N., Thalhammer (2009)).Illustrates the accuracy of time–splitting Hermite and Fourierpseudospectral methods for the GPE
i ∂ t ψ(x , t ) =− 1
2∆ + V (x ) + |ψ(x , t )|2
ψ(x , t )
involving the potential V (x ) =12 (x
21 + x
22 ).
Choose the ground statesolution of the GPEinvolving the potentialV (x ) = x 21 + x 22 as initial
value.Compute a referencesolution at T = 400(128× 128 basis functions,
217
time steps).
−3
−2
−1
0
1
2
3
0
2
4
6
8
10
12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x1
t
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Linear Schrödinger equations Gross–Pitaevskii equationNumerical approximation
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 18/21
Nonlinear Schrödinger equationsNumerical approximationLong-term integration of GPE
Numerical experiment (long-term integration)
Numerical experiment. Illustrates the temporal order of Hermiteand Fourier pseudospectral splitting methods for the GPE
i ∂ t ψ(x , t ) =− 1
2∆ + V (x ) + |ψ(x , t )|2
ψ(x , t ) .
Compute numerical approximations by time-splitting spectralmethods with 2i , 6 ≤ i ≤ 8, basis functions in each spacedirection and 2i , 6 ≤ i ≤ 15, time steps.
Compute numerical approximations by standard explicitRunge–Kutta methods of order four.
Compute error in discrete L2-norm.
Measure the particle number and energy conservation.
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Linear Schrödinger equationsN l S h d
Gross–Pitaevskii equationNumerical approximation
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 19/21
Nonlinear Schrödinger equationsNumerical approximationLong-term integration of GPE
Numerical experiment (long-term integration)
tol. method d.o.f. #transf. ∆pn ∆E
< 10−2 Hermite 2 32 × 32 16384 2.6 · 10−11 4.2 · 10−6
< 10−2 Fourier 2 64 × 64 32768 3.6 · 10−13 1.6 · 10−6
< 10−2 Hermite 4 32 × 32 6144 9.7 · 10−12 1.1 · 10−5
< 10−2 Fourier 4 64 × 64 12288 1.7 · 10−13 9.1 · 10−7
< 10−2 Hermite 6 32 × 32 14337 2.3 · 10−11 3.2 · 10−8
< 10−2 Fourier 6 64 × 64 7169 1.1 · 10−13 6.8 · 10−6
< 10−2 Hermite rk4 32 × 32 65532 2.1 · 10−5 1.2 · 10−4
< 10−2
Fourier rk4 64 × 64 524284 6.4 · 10−10
3.7 · 10−9
< 10−2 Hermite ode45 32 × 32 208376 2.6 · 10−8 1.5 · 10−7
< 10−2 Fourier ode45 64 × 64 1132436 5.6 · 10−12 3.1 · 10−11
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Linear Schrödinger equationsN li S h ödi ti
Gross–Pitaevskii equationNumerical approximation
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
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Nonlinear Schrödinger equationsNumerical approximationLong-term integration of GPE
Observations
The number of required Hermite basis functions resp.transforms are (in general) smaller than the number of Fourier
basis functions resp. transforms.
Splitting methods outperform explicit Runge–Kutta methods.
The splitting methods of order four and six by Blanes andMoan are superior to the second order Strang splitting.
Ch. Neuhauser Convergence of time–splitting pseudospectral methods
Linear Schrödinger equationsNonlinear Schrödinger eq ations
Gross–Pitaevskii equationNumerical approximation
8/3/2019 Ch. Neuhauser and M. Thalhammer- Convergence of high-order time–splitting pseudospectral methods for Schrödinger equations
http://slidepdf.com/reader/full/ch-neuhauser-and-m-thalhammer-convergence-of-high-order-timesplitting 21/21
Nonlinear Schrödinger equationspp
Long-term integration of GPE
Conclusion and future work
Contents. High accuracy discretisations by time-splittingpseudospectral methods.
Convergence analysis for linear evolutionary Schrödingerequations.
Comparison of time-splitting methods regarding accuracy,efficiency, and geometric properties.
Future work.
Provide convergence analysis of high-order time-splitting
pseudospectral methods for nonlinear Schrödinger equations.Study other basis functions, well adapted for a certainpotential.
Ch. Neuhauser Convergence of time–splitting pseudospectral methods