Operator splitting methods for nonlinear Schrödinger equations

47
Nonlinear Schrödinger equations Error analysis of splitting methods Reliable and efficient time integration Operator splitting methods for nonlinear Schrödinger equations Mechthild Thalhammer Leopold–Franzens Universität Innsbruck, Austria Berlin, February 2015 Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Transcript of Operator splitting methods for nonlinear Schrödinger equations

Page 1: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Operator splitting methods fornonlinear Schrödinger equations

Mechthild ThalhammerLeopold–Franzens Universität Innsbruck, Austria

Berlin, February 2015

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 2: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Theme

Splitting methods. Efficient time integration of nonlinear evolutionequations by exponential operator splitting methods

ddt u(t ) = F

(u(t )

)= A(u(t )

)+B(u(t )

), t ∈ (0,T ) ,

u(0) given,

un =SF (τn−1,un−1) =s∏

j=1eas+1− j τn−1DA ebs+1− j τn−1DB un−1

≈ u(tn) = EF(τn−1,u(tn−1)

)= eτn−1DF u(tn−1) , n ∈ 1, . . . , N .

Applications.

Nonlinear Schrödinger equations (GPS, MCTDHF)

Parabolic equations

Wave equations

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 3: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Outline

Theme. Theoretical analysis of exponential operator splitting methodsfor nonlinear Schrödinger equations.

Main inspiration. Work by W. BAO and CH. LUBICH.

Collaborators. Joint work with

W. AUZINGER, H. HOFSTÄTTER, O. KOCH (Vienna)

PH. CHARTIER, F. MÉHATS (Rennes)

S. DESCOMBES (Nice)

Outline.

Applications. Splitting methods for the time integration of the MCTDHF equationsand time-dependent Gross–Pitaevskii equations.

Theoretical analysis. Stability and error analysis of high-order splitting methods forlinear and nonlinear Schrödinger equations.

Practical aspects. Error behaviour of splitting methods for nonlinear Schrödingerequations in different regimes. Reliable and efficient time integration by local errorestimation. Construction and analysis of a posteriori local error estimators forhigh-order splitting methods.

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 4: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

Nonlinear Schrödinger equationsMulti-configuration time-dependent Hartree–Fock equations

Time-dependent Gross–Pitaevskii equations

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 5: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

Electronic Schrödinger equations

Electronic Schrödinger equation. Model for system of unboundfermions interacting by Coulomb force. Time-dependent linearSchrödinger equation for wave function associated with D particlesΨ :R3D × [0,T ] →C : (x, t ) →Ψ(x, t )

i ∂tΨ(x, t ) =H (x)Ψ(x, t ) =AΨ(x, t )+V (x)Ψ(x, t )

subject to asymptotic boundary conditions and initial condition.

Differential operator A comprises second derivatives with respect to each spatialcoordinate

A =− 12 ∆ , ∆=∆x1 +·· ·+∆xD , x = (x1, . . . , xD ) ∈R3D ,

∆xj = ∂2xj 1

+∂2xj 2

+∂2xj 3

, xj = (xj 1, xj 2, xj 3) , j ∈ 1, . . . ,D .

Multiplication operator V takes pairwise Coulomb interaction of particles intoaccount

V (x) = ∑1≤ j<k≤D

1

|xj −xk |, x = (x1, . . . , xD ) ∈R3D .

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 6: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

MCTDHF approach

MCTDHF approach. Multi-configuration time-dependent Hartree–Fockapproach allows to replace high-dimensional linear Schrödinger equationby system of coupled linear ordinary differential equations and nonlinearpartial differential equations in three space dimensions (computationallytreatable problem).

Approximation. Replace multi-particle wave function by linear combination ofHartree products with orbitals depending on coordinates of single particle andsatisfying additional orthogonality and gauge conditions

Ψ(x, t ) ≈ U (x, t ) = ∑µ∈M

aµ(t )Φµ(x, t ) = ∑µ∈M

aµ(t )ϕµ1(x1, t ) · · · ϕµD(xD , t ) .

Pauli exclusion principle. Reduction to antisymmetrised products.

MCTDHF equations. Derivation of equations of motion for orbitals and coefficientsin linear combination of Hartree products via Dirac–Frenkel variational principle

i∂t U −H U ⊥ δU .

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 7: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

Approximation

Approximation. Approximation of multi-particle wave function by linear combination ofHartree products involving orbitals ϕk :R3 × [0,T ] →C for k ∈ 1, . . . ,K

Ψ(x, t ) ≈ U (x, t ) = ∑µ∈M

aµ(t )Φµ(x, t ) ,

M =µ= (µ1, . . . ,µD ) ∈ND :µj ∈ 1, . . . ,K for j ∈ 1, . . . ,D

,

Φµ(x, t ) =( D⊗

j=1ϕµj

)(x, t ) =ϕµ1(x1, t ) · · · ϕµD(xD , t ) , x = (x1, . . . , xD ) ∈R3D , t ∈ (0,T ) .

Pauli exclusion principle. Significant reduction from K D to(K

D

)different coefficients due to

additional requirement of antisymmetry (permutation σ : 1, . . . ,D → 1, . . . ,D)

aσ(µ) = sign(σ) aµ , σ(µ) = (µσ(1), . . . ,µσ(D)

), µ= (µ1, . . . ,µD ) ∈M .

In particular aµ = 0 if µj =µk for some 1 ≤ j < k ≤ D .

Two-particles case involves(K

2)= 1

2 K (K −1) different coefficients: Consideration of σ= (12) with sign(σ) =−1 impliesaµ2µ1 =−aµ1µ2 and in particular aµ1µ1 =−aµ1µ1 = 0 for µ1,µ2 ∈ 1, . . . ,K .

Orthogonality and gauge conditions. Additional orthogonality and gauge conditions onorbitals (with self-adjoint operator A0 =− 1

2∆, t ∈ [0,T ], k,` ∈ 1, . . . ,K )(ϕk (·, t ) |ϕ`(·, t )

)L2 = δk` ,

(ϕk (·, t ) | i ∂tϕ`(·, t )−A0ϕ`(·, t )

)L2 = 0.

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 8: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

MCTDHF equations

MCTDHF equations. Application of Dirac–Frenkel variational principle yields associatedequations of motion. MCTDHF equations constitute system of ordinary differentialequations for coefficients aµ : [0,T ] →C (µ ∈M ) and partial differential equations for

orbitals ϕk :R3 × [0,T ] →C (k ∈ 1, . . . ,K )

i ddt aµ(t ) = ∑

ν∈M

(Φµ(·, t ) |V Φν(·, t )

)L2 aν(t ) ,

i ∂tϕk (ξ, t ) =A0ϕk (ξ, t )+ (I −P )K∑

`,m=1ρ−1

k` (t )(Ψ`(·, t ) |V Ψm (·, t )

)L2 ϕm (ξ, t ) ,

ξ ∈R3 , t ∈ [0,T ] , k ∈ 1, . . . ,K , µ ∈M ,

subject to asymptotic boundary conditions and initial conditions.

Nonlinearity. Nonlinearity comprises terms of the form∫R3

1

|ξ−η| f1(ξ) g1(ξ) dξ f2(η) , η ∈R3 ,ÏR3×R3

1

|ξ−η| f1(ξ) f2(η) g1(ξ) g2(η) dξdη .

Remark. Second iterated commutator of Laplacian and nonlinearity comprises certaincombinations of orbitals

f1 f2∆ f3 , f3 ∇ f1 ·∇ f3 .

Essential for error analysis of splitting methods (considerations for MCTDHF equations inprinciple similar to considerations for cubic Schrödinger equation).

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 9: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

Existence and regularity result

Existence result. Theoretial result ensuring existence and uniqueness ofregular solution to MCTDHF equations essential for stability andconvergence analysis of splitting methods.

Theorem (Koch and Lubich, 2008)

Let m ∈N≥2. Provided that the initial density matrix ρ(0) is non-singularand that the initial values for the orbitals satisfy ϕk (·,0) ∈ H m(R3) fork ∈ 1, . . . ,K , there exists a unique strong solution to the MCTDHFequations such that

ϕk (·, t ) ∈ H m(R3) , k ∈ 1, . . . ,K , t ∈ [0,T ) ,

where either T =∞ or ρ(t ) becomes singular for t ↑ T .

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 10: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

Bose–Einstein condensation

In our laboratories temperatures aremeasured in micro- or nanokelvin ...In this ultracold world ... atoms moveat a snail’s pace ... and behave likematter waves. Interesting andfascinating new states of quantummatter are formed and investigated inour experiments.

(GRIMM ET AL., Innsbruck)

Bose–Einstein condensation in dilute gases. In 1925 Albert Einstein predicted that at (very)low temperatures particles in a (dilute) gas could all reside in the same quantum state. Thispeculiar gaseous state, a Bose–Einstein condensate, was produced in the laboratory for thefirst time in 1995 using the powerful laser-cooling methods developed in recent years. Thesecondensates exhibit quantum phenomena on a large scale, and investigating them hasbecome one of the most active areas of research in contemporary physics. See PETHICK,SMITH (2002).

Physical experiments (University of Innsbruck). Realisation of groundstate and investigation of time evolution (H.-C. NÄGERL, M. MARK).

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 11: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

Gross–Pitaevskii equations

Physical experiments. Observation of multi-componentBose–Einstein condensates. Realisation of double species87Rb 41K BEC at LENS, see G. THALHAMMER ET AL. (2008).

Theoretical model. Mathematical description (of certain aspects) ofmulti-component Bose–Einstein condensate by system of coupledtime-dependent Gross–Pitaevskii equations for macroscopic wavefunctionΨ :Rd × [0,T ] →CJ : (x, t ) 7→Ψ(x, t )

i ħ ∂tΨj (x, t ) =(− ħ2

2mj∆+Vj (x)+ħ2

J∑k=1

gjk∣∣Ψk (x, t )

∣∣2)Ψj (x, t ) ,

Vj (x) ≈d∑`=1

(mj

2 ω2j` (x`−ζj`)2 +κj`

(sin(qj`x`)

)2)

,∥∥Ψj (·,0)

∥∥2L2 = Nj ,

x ∈Rd , t ∈ [0,T ] , j ∈ 1, . . . , J .

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 12: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

Model equation

Model problem. Consider nonlinear Schrödingerequation for ψ :Rd × [0,T ] →C : (x, t ) 7→ψ(x, t )

i ∂tψ(x, t ) =(− 1

2 ∆+U (x)+ϑ ∣∣ψ(x, t )∣∣2

)ψ(x, t ) ,

ψ(x,0) given, (x, t ) ∈Rd × [0,T ] ,

subject to asymptotic boundary conditions. −10

1

−10

1

−1

0

1

x2x1

x3

Illustration. Solution profile |ψ|2 of GPE in 3D (ω=ϑ= 1, T = 3, M = 1283, tol = 10−6).

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

E(ψ(·, t )

)= ((− 12 ∆+U + 1

2 ϑ |ψ(·, t )|2)ψ(·, t )

∣∣∣ψ(·, t ))

L2 , t ∈ [0,T ] .

Ground state. Solution of special form ψ(x, t ) = e− iµt ϕ(x) that minimises energyfunctional. Useful as reliable reference solution in time integration.

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 13: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

Illustration (Ground state computation, Time evolution)

Simulation. Model equation with harmonic potential (d = 2, ϑ= 10). Computation ofgroundstate solution (ω1 = 1 =ω2) and time evolution (ω1 = 2 =ω2) for

i ∂tψ(x, t ) =(− 1

2 ∆+U (x)+ϑ |ψ(x, t )|2)ψ(x, t ) .

Space discretisation by Fourier pseudo-spectral method (x ∈ [−8,8]× [−8,8], M = 200×200).Artificial time integration by 2(1) pair based on Strang and Lie–Trotter splitting methods.Time integration by embedded 4(3) pair based on 4th-order splitting method by BLANES,MOAN (2002) (t ∈ [0,4], tol = 10−6).

MovieGround state, Time Evolution, Energy, Time stepsizes (MATLAB)

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 14: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

Illustrations (Ground state computation, Time evolution)

20 40 60 80 100 120 140 1601.5

2

2.5

3Energy versus step number

20 40 60 80 100 120 140

0.005

0.01

0.015

0.02

Time stepsize versus step number

20 40 60 80 100 120 140 160

2.3883

2.3883

2.3883

2.3883

2.3883

2.3883

2.3883

2.3883

2.3883

2.3883

Energy versus step number

20 40 60 80 100 120 140

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Time stepsize versus step number

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 15: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

Abstract formulation

Abstract formulation. Compact formulation of model equation asnonlinear evolution equation on Hilbert space (X , (· | ·)X ,‖ ·‖X )

ddt u(t ) = F

(u(t )

)= A u(t )+B(u(t )

)u(t ) , t ∈ (0,T ) .

Linear subproblem. Solution representation for subproblem involving lineardifferential operator A (related to Laplacian) often relies on spectral decomposition(employ eigenrelation A Bm =µm Bm for m ∈M )

ddt v(t ) = A v(t ) , t ∈ (0,T ) ,

v(0) = ∑m∈M

vm Bm , v(t ) = et A v(0) = ∑m∈M

vm et µm Bm , t ∈ [0,T ] .

Nonlinear subproblem. Special invariance property of solution to subprobleminvolving (unbounded) nonlinear multiplication operator B (related to potential andnonlinearity) allows to compute solution by pointwise multiplication

ddt w(t ) = B

(w(t )

)w(t ) = B(w(0)) w(t ) , t ∈ (0,T ) ,(

w(t ))(x) = (

etB(w(0))w(0))(x) = et (B(w0))(x)w0(x) , (x, t ) ∈Rd × [0,T ] .

Explanation. Analytical solution of ∂tψ(x, t ) =− i(U (x)+ϑ ∣∣ψ(x, t )

∣∣2)ψ(x, t ) satisfies relation

∂t∣∣ψ(x, t )

∣∣2 = ∂t(ψ(x, t )ψ(x, t )

)= 2ℜ(ψ(x, t ) ∂tψ(x, t )

)= 2ℜ(− i(U (x)+ϑ ∣∣ψ(x, t )

∣∣2) |ψ(x, t )|2)= 0.

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 16: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

Highly oscillatory problems, Semi-classical regime

Additional small parameter 0 < ε<< 1.

Highly oscillatory equations. Consider highly oscillatory equation

ddt u(t ) = 1

ε A u(t )+B(u(t )

)u(t ) , t ∈ (0,T ) .

Rescaling leads to problem over long times

ddt u(t ) = A u(t )+εB

(u(t )

)u(t ) , t ∈ (

0, Tε

).

Semi-classical regime. Consider nonlinear Schrödinger equation

ddt u(t ) = εA u(t )+ 1

ε B(u(t )

)u(t ) , t ∈ (0,T ) .

Problems of similar form arise in applications from solid state physics, seeBAO, JIN, MARKOWICH (2002/03).

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 17: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

MCTDHF equationsGross–Pitaevskii equationsModel equation, Illustrations

Illustration (Small parameter, Solution behaviour)

Simulation. Consider model equation with harmonic potential and additional smallparameter (d = 1, ε= 10−2, ω= 1, ϑ= 1)

i ∂tψ(x, t ) =(− 1

2 ε∆+ 1ε U (x)+ 1

ε ϑ |ψ(x, t )|2)ψ(x, t ) .

Space and time discretisation by Fourier pseudo-spectral method and embedded 4(3) pairbased on 4th-order splitting method by BLANES, MOAN (2002) (x ∈ [−8,8], M = 8192,t ∈ [0,3], tol = 10−6, N = 2178).

MovieSolution behaviour in presence of small parameter (MATLAB)

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 18: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Exponential operator splitting methodsfor nonlinear Schrödinger equations

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 19: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Exponential operator splitting methods

Time-stepping approach. For nonlinear evolution equation on Banach space (X ,‖ ·‖X )

ddt u(t ) = F

(u(t )

)= A(u(t )

)+B(u(t )

), t ∈ (0,T ) , u(0) given,

determine approximations at time grid points 0 = t0 < ·· · < tN ≤ T with associated stepsizesτn−1 = tn − tn−1 for n ∈ 1, . . . , N by recurrence

un =SF (τn−1,un−1) ≈ u(tn ) = EF(τn−1,u(tn−1)

)= eτn−1DF u(tn−1) .

Splitting methods. Time-splitting methods rely on suitabledecomposition of right-hand side and presumption that subproblems

ddt v(t ) = A

(v(t )

), v(t ) = etDA v(0) , t ∈ (0,T ) ,

ddt w(t ) = B

(w(t )

), w(t ) = etDB w(0) , t ∈ (0,T ) ,

are solvable in accurate and efficient manner.

General form. High-order splitting methods are cast into following form with real (orcomplex) method coefficients (a j ,b j )s

j=1

SF (t , ·) =s∏

j=1eas+1− j tDA ebs+1− j tDB ≈ EF (t , ·) = etDF = et (DA+DB ) .

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 20: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Calculus of Lie-derivatives

Formal calculus. Calculus of Lie-derivatives is suggestive of less involved linear case, see forinstance HAIRER, LUBICH, WANNER (2002) and SANZ-SERNA, CALVO (1994).

Problem. Consider nonlinear evolution equation on Banach space involving (unbounded)nonlinear operator F : D(F ) ⊆ X → X

ddt u(t ) = F

(u(t )

), t ∈ (0,T ) .

Employ formal notation for exact solution

u(t ) = EF(t ,u(0)

)= e tDF u(0) , t ∈ [0,T ] .

Evolution operator, Lie-derivative. For (unbounded) nonlinear operator G : D(G) ⊆ X → Xdefine evolution operator and Lie-derivative by

e tDF G v =G(EF (t , v)

), t ∈ (0,T ) , DF G v =G ′(v)F (v) .

Remark. Definition of Lie-derivative is natural extension of identity L = ddt

∣∣∣t=0

e tL

ddt

∣∣∣t=0

e tDF G v = ddt

∣∣∣t=0

G(EF (t , v)

)=G ′(EF (t , v))

F(EF (t , v)

)∣∣∣t=0

=G ′(v)F (v)

= DF G v .

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 21: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Example methods

General form. High-order splitting methods are cast into following form with real (orcomplex) method coefficients (a j ,b j )s

j=1

SF (t , ·) =s∏

j=1eas+1− j tDA ebs+1− j tDB ≈ EF (t , ·) = etDF = et (D A+DB ) .

Low-order methods. First-order Lie–Trotter splitting method andsecond-order Strang splitting method

SF (t , ·) = etDB etDA , SF (t , ·) = e12 tDA etDB e

12 tDA .

Higher-order method. Symmetric fourth-order splitting methodby BLANES, MOAN (2002).

j aj

1 02,7 0.2452989571842713,6 0.6048726657110804,5 1/2− (a2 +a3)

j bj

1,7 0.08298440641740522,6 0.39630980149836803,5 −0.03905630492234864 1−2(b1 +b2 +b3)

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 22: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Practical realisation by pseudo-spectral methods

Spectral decomposition. Numerical solution of subproblem involving linear differentialoperator A (related to Laplacian) relies on spectral decomposition (employ eigenrelationA Bm =µm Bm for m ∈M )

ddt v(t ) = A v(t ) , t ∈ (0,T ) , v(0) given,

v(0) = ∑m∈M

vm Bm , v(t ) = et A v(0) = ∑m∈M

vm et µm Bm , t ∈ [0,T ] .

Fourier spectral decomposition. LetΩ= (−a1, a1)×·· ·× (−ad , ad ) (sufficiently large).Fourier basis functions form complete orthonormal system in L2(Ω) and satisfy

ψ(·, t ) = ∑m∈M

ψm (t )Fm , ψm (t ) = (ψ(·, t ) |Fm

)L2 ,

−∆Fm =λm Fm , Fm (x) =d∏`=1

eiπm`

( x`a`

+1)

p2a`

, λm =d∑`=1

π2m2`

a2`

.

Numerical approximation. Truncation of infinite sum and application of trapezoidquadrature formula yields approximation (realisation by Fast Fourier Techniques)

QM ψ(·, t ) = ∑|m|≤M

ψm (t )Fm , ψm (t ) =∫Ωψ(x, t )Fm (x) dx ≈ ∑

k∈K

ωk ψ(ξk , t )Fm (ξk ) .

Remark. Analogous relations for Sine, Hermite, and generalised Laguerre–Fourier Hermitepseudo-spectral methods.

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 23: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Illustration (GPE with rotation, Time evolution)

Simulation. Time-dependent Gross–Pitaevskii equation with additional rotation term(EXAMPLE IN BAO, LI, SHEN, 2009).

MovieTime evolution of GPE with rotation term (MATLAB)

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 24: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Stability and error analysis ofexponential operator splitting methods

for nonlinear Schrödinger equations

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 25: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Objectives

Mein Verzicht auf das Restglied war leichtsinnig. (W. ROMBERG, 1979)

Situation. Time integration of nonlinear evolution equations by splitting methods

ddt u(t ) = F

(u(t )

)= A(u(t )

)+B(u(t )

), t ∈ (0,T ) , u(0) given,

un =SF (τn−1,un−1) =s∏

j=1eas+1− j τn−1DA ebs+1− j τn−1DB un−1

≈ u(tn ) = EF(τn−1,u(tn−1)

)= eτn−1DF u(tn−1) , n ∈ 1, . . . , N .

Reasonable requirements in connection with nonlinear Schrödinger equations (regularsolution, WKB initial condition, with k ∈N)

supt∈[0,T ]

‖u(t )‖D ≤C , ε j ‖∂kx u(0)‖X ≤C .

Local error representations. Deduce local error representations for high-order splittingmethods suitable for problems involving unbounded operators and critical parameters

LF (τ, v) =SF (τ, v)−EF (τ, v) =O(τp+1,‖v‖D

).

Convergence analysis. Derivation of convergence result relies on stability bounds and localerror estimates. Extension to full discretisations by time-splitting pseudo-spectral methods

‖uN M −u(tN )‖X ≤C(‖u0 −u(0)‖X +τp +M−q

).

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 26: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Derivation of local error expansions

Standard approaches.

Expansion of exponential functions

Baker–Campbell–Hausdorff formula

Approaches for nonlinear evolution equations.

Quadrature formulas. Obtain optimal error bounds with respect toregularity of solution, see JAHNKE, LUBICH (2000), LUBICH (2008).Extension to high-order splitting methods TH. (2008, 2012), KOCH, NEUHAUSER,TH. (2013), HOFSTÄTTER, KOCH, TH. (2014).

Differential equations. Obtain exact local error representations forevolution equations involving critical parameters, see DESCOMBES,SCHATZMAN (2002), DESCOMBES, DUMONT, LOUVET, MASSOT (2007).Application to Schrödinger equations in semi-classical regime, see DESCOMBES,TH. (2010, 2012). Related techniques studied in AUZINGER, KOCH, TH. (2012, 2014).

Highly oscillatory problems. Recent study of multi-revolution composition time-splittingpseudo-spectral methods for highly oscillatory problems (with CHARTIER, MÉHATS).

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 27: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Approach based on quadrature formulas

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 28: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Quadrature formulas

Approach. Suitable local error expansions for high-order splittingmethods allow to deduce error estimates for linear and nonlinearSchrödinger equations. Numerical experiments confirm optimalitywith respect to regularity of exact solution.

LF (t , v) =SF (t , v)−EF (t , v) =O(t p+1,‖v‖D

)Guiding principle. Linear problems versus nonlinear problems

D = H p (Ω) , D = H2p (Ω) .

Numerical experiments confirm order reduction for less regular solutions.

Main tools.

Variation-of-constants formula(Gröbner–Alekseev)

Stepwise expansion of etDB

Quadrature formulas for multipleintegrals

Bounds for iteratedLie-commutators

Characterise domains ofunbounded operators

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 29: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Local error expansion (Linear evolution equations)

Situation. Splitting method involving two compositions applied to linear evolution equation

ddt u(t ) = A u(t )+B u(t ) , SF (t , ·) = eb2tB ea2t A eb1tB ea1t A ≈ EF (t , ·) = et (A+B) .

Derivation of local error expansion. Assume that ‖B‖X←X ≤CB and ‖etC ‖X←X ≤ eMC t .Variation-of-constants formula yields suitable representation for exact solution

EF (t , ·) = et A + I1 + I2 +O(t 3,C 3

B , MA , MA+B)

,

I1 =∫ t

0e(h−τ1)A B eτ1 A dτ1 , I2 =

∫ t

0

∫ τ1

0e(t−τ1)A B e(τ1−τ2)A B eτ2 A dτ2 dτ1 .

Stepwise expansion of etB yields suitable representation for numerical solution

SF (t , ·) = et (a1+a2)A +Q1 +Q2 +O(t 3,C 3

B , MA , MB)

,

Q1 = t(b1 ea2t A B ea1t A +b2 B et (a1+a2)A)

,

Q2 = 12 t 2 (

b21 ea2t A B2ea1t A +2b1b2 B ea2t A B ea1t A +b2

2 B2 et (a1+a2)A).

Under basic consistency condition a1 +a2 = 1 obtain local error expansion involvingmultiple integrals I1, I2, and iterated quadrature approximations Q1,Q2

LF (t , ·) =Q1 − I1 +Q2 − I2 +O(t 3,C 3

B , MA , MB , MA+B)

.

Taylor series expansions of integrands involve iterated Lie-commutators [A,B ], [A, [A,B ]].

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 30: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Local error expansion (Linear evolution equations)

Assumptions. Assume a1 +a2 = 1 and furthermore

‖B‖X←X ≤CB , ‖etC ‖X←X ≤ eMC t , C ∈ A,B , A+B ,

‖[A,B ]v‖X +‖[A, [A,B ]]v‖X ≤Cad ‖v‖D .

Local error expansion. Splitting method involving two compositions fulfills local errorexpansion (Strang a1 = 1

2 = a2, b1 = 1, b2 = 0)

LF (t , v) = (eb2tB ea2t A eb1tB ea1t A −et (A+B))v

= t(b1 +b2 −1

)et A B v

− t 2 et A((

a1b1 +b2 − 12

)[A,B

]+ 12

((b1 +b2)2 −1

)B2

)v

+O(t 3,C 3

B , MA , MB , MA+B ,Cad,‖v‖D)

.

Applications (Schrödinger equations). Extension to high-order splittingmethods for nonlinear Schrödinger equations (GPE), see TH. (2008, 2012).

Natural choice X = L2(Ω) ensures MA = MB = MA+B = 0 (unitarity of evolutionoperators). Iterated Lie-commutators related to differential operators. Explainsregularity requirements D = H2p (Ω) (nonlinear case) and D = H p (Ω) (linear case).

Drawback. Numerical illustrations show that approach not optimal with respect tocritical parameter (B =U /ε).

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 31: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Local error expansion (Nonlinear evolution equations)

Result. Local error expansion for high-order splitting methods applied to linear andnonlinear evolution equations.

Theorem (Th. 2008, Th. 2012, Koch & Neuhauser & Th. 2013)

Any exponential operator splitting method of (nonstiff) order p admits the(formal) expansion

LF (t , ·) =p∑

k=1

∑µ∈Nk

|µ|≤p−k

1µ! t k+|µ| Ckµ

k∏`=1

adµ`D A

(DB ) etD A +Rp+1(t , ·) ,

Ckµ =∑λ∈Λk

αλ

k∏`=1

bλ` cµ`λ`

−k∏`=1

1µ`+···+µk+k−`+1 .

Remarks.

Application to Gross–Pitaevskii equations and MCTDHF equations.

Local error expansion suitable for parabolic problems.

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 32: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Convergence result (MCTDHF equations)

Situation and assumptions.

Consider an exponential operator splitting method of (nonstiff) order p ≥ 1.

Set m = p = 2 or m = 2 p −3 for p ∈N≥3.

Suppose that the MCTDHF equations possess a uniquely determined sufficientlyregular solution on the time interval [0,T ] such that

‖u(t )‖Xm ≤ Mm , u(t ) = (a(t ),ϕ(·, t )

)T , Xm =C|M |×(Hm(

R3))K , t ∈ [0,T ] .

Assume that the associated density matrix ρ(t ) is nonsingular for t ∈ [0,T ].

Theorem (Koch & Neuhauser & Th., 2013)

The global error estimate

‖un −u(tn )‖X0≤C τp , n ∈ 0, . . . , N , tN ≤ T ,

holds with constant C > 0 depending on Mm .

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 33: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Global error estimate (Full discretisations)

Discretisation. Space and time discretisation of nonlinear Schrödinger equations (GPE) bydifferent pseudo-spectral methods (Fourier, Sine, Hermite) and high-order variable stepsizetime-splitting methods.

Theorem (Th. 2012)

Provided that exact solution remains bounded in fractional power space Xβ

defined by principal linear part for β≥ p, the global error estimate holds∥∥uN M −u(tN )∥∥

X0≤C

(∥∥u0 −u(0)∥∥

X0+τp

max +M−q)

.

Extensions.

Time-dependent Gross–Pitaevskii equations with additional rotation term, seeHOFSTÄTTER, KOCH, TH. (2014).

Multi-revolution composition time-splitting pseudo-spectral methods for highlyoscillatory problems (with CHARTIER, MÉHATS).

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 34: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Global error estimate (Full discretisations)

Theorem (Th. 2012)

Global error estimate for regular solutions∥∥uN M −u(tN )∥∥

X0≤C

(∥∥u0 −u(0)∥∥

X0+τp

max +M−q)

.

Illustration. Discretisation of Gross–Pitaevskii equation (d = 2, ε=ω= T = 1) by differentpseudo-spectral methods (M = 256×256) and time-splitting methods of (nonstiff) ordersp = 1,2,3,4. Dependence of global error on total number of basis functions (ϑ= 0, dominanterror term related to linear part, Fourier, Hermite basis function as exact reference solution,temporal error dominates global error). Numerically observed orders of convergence in time(ϑ= 1, Fourier, Sine, Hermite, smooth initial value, numerical reference solution).

102

103

104

105

10−15

10−10

10−5

100

total number of basis functions

glob

al e

rror

order 1order 2order 3order 4

10−3

10−2

10−1

10010

−10

10−5

100

time stepsize

glob

al e

rror

order 1order 2order 3order 4

10−3

10−2

10−1

10010

−10

10−5

100

time stepsize

glob

al e

rror

order 1order 2order 3order 4

10−3

10−2

10−1

10010

−10

10−5

100

time stepsize

glob

al e

rror

order 1order 2order 3order 4

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 35: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Approach based on differential equations

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 36: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Differential equations

Approach. Derivation of compact local error representation for splitting methods applied to(non)linear evolution equations involving critical parameters. Related approach studied forconstruction and analysis of a posteriori local error estimators (with W. AUZINGER, O. KOCH).

Basic idea. Deduce differential equation for splitting operator that is closely related todifferential equation for evolution operator

SF (t , ·) =s∏

j=1eas+1− j tD A ebs+1− j tDB , t ∈ (0,T ) , SF (0, ·) = I ,

ddt EF (t , ·) = DF EF (t , ·) = (D A +DB )EF (t , ·) , t ∈ (0,T ) , EF (0, ·) = I .

Main tools. Variation-of-constants formula, iterated commutators.

Theorem (Descombes & Th. 2012)

The defect of the Lie–Trotter splitting method admits the integral representation

LF (t , ·) =∫ t

0

∫ τ1

0eτ1D A eτ2DB

[D A ,DB

]e(τ1−τ2)DB e(t−τ1)DF dτ2 dτ1

=∫ t

0

∫ τ1

0∂2EF

(t −τ1,SF (τ1, ·)) ∂2EB

(τ1 −τ2,EA (τ1, ·))[B , A

](EB

(τ2,EA (τ1, ·))) dτ2 dτ1 .

Remark. Extension to higher-order splitting methods involved! Corresponding result forhigh-order splitting methods applied to linear problems deduced in DESCOMBES, TH. (2010).

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 37: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Splitting methodsConvergence analysis

Application (Equations with critical parameters)

Application. Error analysis of splitting methods for Schrödingerequations involving critical parameters 0 < ε<< 1

i ∂tψ(x, t ) =(− 1

2 ε∆+ 1ε U (x)+ 1

ε ϑ∣∣ψ(x, t )

∣∣2)ψ(x, t ) , t ∈ (0,T ) ,

see DESCOMBES, TH. (2010, 2012).

High-order splitting methods for linear Schrödinger equations

local error = O( τp+1

ε

).

Lie–Trotter splitting method for nonlinear evolution equations

smooth initial value: local error = C( τε

)τ2 ,

WKB initial value: local error = C( τε

)τ .

Remark. Difficult task to adjust time stepsize in suitable manner. Reliable time integrationof Schrödinger equations with critical parameters based on adaptive time stepsize control.

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 38: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Local error estimatorsIllustrations (Smaller parameter)

Reliable and efficient time integrationof nonlinear Schrödinger equations

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 39: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Local error estimatorsIllustrations (Smaller parameter)

Adaptive stepsize control

Adaptive stepsize control. Standard strategy for adaptive time stepsize control

τoptimal = τ ·min

(αmax,max

(αmin, p+1

√α · tol

errlocal

)).

Local error estimators. Construction of local error estimators for splitting methods.

A posteriori local error estimators, see AUZINGER, KOCH, TH. (2012, 2014).

Embedded splitting methods, see KOCH, NEUHAUSER, TH. (2013).

Example. Fourth-order splitting method (BLANES, MOAN) and embedded third-ordersplitting method (KOCH, TH.).

j a j

1 02,7 0.2452989571842713,6 0.6048726657110804,5 1/2− (a2 +a3)

j aj

1 a12 a23 a34 a45 0.37521626932368286 1.48786665947379467 −1.3630829287974774

j b j

1,7 0.08298440641740522,6 0.39630980149836803,5 −0.03905630492234864 1−2(b1 +b2 +b3)

j bj

1 b12 b23 b34 b45 0.44633743544204996 −0.00609953244862537 0

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 40: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Local error estimatorsIllustrations (Smaller parameter)

A posteriori error estimators (Linear evolution equations)

Basic idea. Consider linear evolution equation

ddt EF (t ) = (

A+B)EF (t ) , t ∈ [0,T ] , EF (0) = I .

For p-th order splitting method determine related differential equation involving defect

ddt SF (t ) = (

A+B)SF (t )+DF (t ) , t ∈ [0,T ] , SF (0) = I .

Use integral representation for local error LF =SF −EF (variation-of-constants formula)

ddt LF (t ) = (

A+B)LF (t )+DF (t ) , t ∈ [0,T ] , LF (0) = 0,

LF (t ) =∫ t

0e(t−τ)(A+B) DF (τ) dτ .

Apply Hermite quadrature approximation and show that f (0) = ·· · = f (p−1)(0) = 0 due tovalidity of order conditions

p−1∑`=0

ω`t`+1 f (`)(0)+ 1p+1 t f (t )−

∫ t

0f (τ) dτ=O

(t p+2)

.

Obtain asymptotically correct a posteriori local error estimator

PF (t ) = 1p+1 t DF (t ) , PF (t )−LF (t ) =O

(t p+2)

.

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 41: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Local error estimatorsIllustrations (Smaller parameter)

A posteriori local error estimators

Result. Asymptotically correct a posteriori local error estimatorassociated with high-order splitting method

SF =S s1 ≈ EF , S m

k (t ) =m∏

j=kebj tB eaj t A ,

PF = 1p+1 t DF , PF (t )−LF (t ) =O

(t p+2) ,

DF =s∑

k=1S s

k ak A S k−11 +

s−1∑k=1

S sk+1 bk B S k

1 − (A+ (1−bs )B

)SF (t ) .

Extension to nonlinear evolution equations by calculus of Lie-derivatives.

Rigorous analysis. Detailed treatment of high-order splitting methods for nonlinearevolution equations involved! Theoretical analysis completed for Strang splitting methodand linear equations, see AUZINGER, KOCH, TH. (2012, 2014).

Suitable representations for higher-order derivatives of defect.

Rigorous proof of asymptotical correctness in context of Schrödinger equation(optimal regularity requirements on exact solution).

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 42: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Local error estimatorsIllustrations (Smaller parameter)

A posteriori error estimator (Lie–Trotter splitting method)

Special case. A posteriori local error estimator for Lie–Trotter splitting method applied tolinear evolution equation given by

PF (t , v) = 12 t DF (t , v) , DF (t , v) = (

etB et A A− A etB et A)v .

Extension to nonlinear case yields

DF (t , v) = ∂2EB(t ,EA (t , v)

)∂2EA (t , v) A v − A EB

(t ,EA (t , v)

).

Explanation. Extension by formal calculus of Lie-derivatives implies DF (t , v) = D A etD A etDB v −etD A etDB D A v and

G(v) = etD A etDB v = EB(t ,EA (t , v)

), G′(v) = ∂2EB

(t ,EA (t , v)

)∂2EA (t , v) ,

etD A etDB D A v = A EB(t ,EA (t , v)

), D A etD A etDB v =G′(v) A v = ∂2 EB

(t ,EA (t , v)

)∂2EA (t , v) A v .

Remark. Improved approximation SF (t , ·)−PF (t , ·) = EF (t , ·)+O(t p+2)

.

Realisation and computational effort. Realisation for nonlinear Schrödinger equations(Gross–Pitaevskii equation) straightforward. Computational effort comparable with splittingpair Lie/Strang (two additional applications of A required, FFT).

P (t , v) = e− it (U+ϑ |w |2)(

A w − iϑt(

A w |w |2 + A w w2))− A e− it (U+ϑ |w |2)w , w = et A v .

Explanation. With G(v) = e− it (U+ϑ |w |2)w , G′(v) = e− it (U+ϑ |w |2)(et A (·)− iϑt

(w et A (·)+w et A (·))w

)obtain

etD A etDB D A v = A e− it (U+ϑ |w |2)w , D A etD A etDB v = e− it (U+ϑ |w |2)(

A w − iϑ t(

Aw |w |2 + A w w2)).

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 43: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Local error estimatorsIllustrations (Smaller parameter)

Illustration (Smaller parameter)

Integration without preparation is frustration. (REVEREND LEON SULLIVAN, )

Situation. Time and space discretisation of model problem (d = 1, ε= 10−2, ϑ= 1,(x, t ) ∈ [−8,8]× [0,3]) by splitting methods and Fourier pseudo-spectral method (M = 8192).

Illustration. Harmonic potential with ω= 1 (col. 1 & 2) or ω= 2 (col. 3 & 4), respectively.Splitting methods (p = 1,4) applied with constant time stepsizes. Comparison of solutionprofiles |ψ(x, t )|2 for x ∈ [0,1.5] at time t = 3. Stepsizes τ= ε

20 (col. 1 & 3) or τ= ε50 (col. 2 & 4)

if p = 1, and τ= ε20 if p = 4.

0 0.5 1 1.50

0.2

0.4

0.6

0.8

x

p=4p=1

0 0.5 1 1.50

0.2

0.4

0.6

0.8

x

p=4p=1

0 0.5 1 1.50

0.2

0.4

0.6

0.8

x

p=4p=1

0 0.5 1 1.50

0.2

0.4

0.6

0.8

x

p=4p=1

MovieTime integration by embedded 4(3) splitting pair. Solution profile |ψ(x, t )|2 for

tol = 10−1,10−2,10−3,10−6 (N = 951,2342,2452,3560). (MATLAB)

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 44: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Local error estimatorsIllustrations (Smaller parameter)

Illustration (Reliable time integration)

Further illustrations. Time integration of model equation (d = 1, ε= 1, ω= 5) by embedded4(3) pair (tol = 10−10). Solution profiles ℜψ for (x, t ) ∈ [0,1.5]× [T0,T ] and associated timestepsizes. Left: Additional lattice potential with κ= 10 and defocusing nonlinearity withϑ= 1 for t ∈ [0,10]. Middle: Focusing nonlinearity with ϑ=−10 for t ∈ [0,1]. Right:Defocusing nonlinearity with ϑ= 1 and sharp initial Gaussian with γ= 4 for t ∈ [0,10].

1 102470.87

1.1439x 10

−3

step number

time

step

1 70440.7989

7.6453x 10

−4

step number

time

step

1 116253.5993

16.2679x 10

−4

step number

time

step

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 45: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Local error estimatorsIllustrations (Smaller parameter)

Conclusions and future work

Conclusions.

Theoretical analysis of space and time discretisations based spectraland splitting methods.

Adaptivity in time essential for reliable numerical simulations.

Open questions.

Asymptotical correctness of higher-order a posteriori local error estimators fornonlinear Schrödinger equations.

Convergence analysis of higher-order time-splitting pseudo-spectral methods fornonlinear Schrödinger equations involving small parameters iu′ = Au + 1

εB(u).

Convergence analysis of multi-revolution compositon methods combined withtime-splitting pseudo-spectral methods for Schrödinger equations iu′ = 1

ε Au +B(u).

Thank you!

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 46: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Local error estimatorsIllustrations (Smaller parameter)

References

Publications.

1. W. AUZINGER, O. KOCH, M. TH. Defect-based local error estimators for splitting methods, with application toSchrödinger equations. Part I. The linear case. J. Comput. Appl. Math. 236 (2012) 2643–2659.

2. W. AUZINGER, O. KOCH, M. TH. Defect-based local error estimators for splitting methods, with application toSchrödinger equations. Part II. Higher-order methods for linear problems. J. Comput. Appl. Math. 255 (2014) 384–403.

3. M. CALIARI, CH. NEUHAUSER, M. TH. High-order time-splitting Hermite and Fourier spectral methods for theGross–Pitaevskii equation. J. Comput. Phys. 228 (2009) 822–832.

4. M. CALIARI, A. OSTERMANN, S. RAINER, M. TH. A minimisation approach for computing the ground state ofGross–Pitaevskii systems. J. Comput. Phys. 228 (2009) 349–360.

5. S. DESCOMBES, M. TH. An exact local error representation of exponential operator splitting methods for evolutionaryproblems and applications to linear Schrödinger equations in the semi-classical regime. BIT 50 (2010) 729–749.

6. S. DESCOMBES, M. TH. The Lie–Trotter splitting for nonlinear evolutionary problems with critical parameters. Acompact local error representation and application to nonlinear Schrödinger equations in the semi-classical regime.IMA J. Numer. Anal. 33/2 (2013) 722–745.

7. H. HOFSTÄTTER, O. KOCH, M. TH. Convergence analysis of high-order time-splitting pseudo-spectral methods forGross–Pitaevskii equations with rotation term. Numer. Math., doi 10.1007/s00211-013-0586-9.

8. O. KOCH, CH. NEUHAUSER, M. TH. Error analysis of high-order splitting methods for nonlinear evolutionarySchrödinger equations and application to the MCTDHF equations in electron dynamics. M2AN 47/5 (2013) 1265–1286.

9. O. KOCH, CH. NEUHAUSER, M. TH. Embedded exponential operator splitting methods for the time integration ofnonlinear evolution equations. Appl. Numer. Math. 63 (2013) 14–24.

10. CH. NEUHAUSER, M. TH. On the convergence of splitting methods for linear evolutionary Schrödinger equationsinvolving an unbounded potential. BIT 49 (2009) 199–215.

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations

Page 47: Operator splitting methods for nonlinear Schrödinger equations

Nonlinear Schrödinger equationsError analysis of splitting methods

Reliable and efficient time integration

Local error estimatorsIllustrations (Smaller parameter)

References

11. M. TH. High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM J. Numer.Anal. 46/4 (2008) 2022–2038.

12. M. TH. Convergence analysis of high-order time-splitting pseudo-spectral methods for nonlinear Schrödinger equations.SIAM J. Numer. Anal. 50/6 (2012) 3231–3258.

13. M. TH., J. ABHAU. A numerical study of adaptive space and time discretisations for Gross–Pitaevskii equations.J. Comp. Phys. 231 (2012) 6665–6681.

Submitted manuscripts.

1. W. AUZINGER, H. HOFSTÄTTER, O. KOCH, M. TH. Defect-based local error estimators for splitting methods, withapplication to Schrödinger equations. Part III. The nonlinear case.

2. P. CHARTIER, F. MÉHATS, M. THALHAMMER, Y. ZHANG. Superconvergence of Strang splitting for NLS in T d .

Manuscripts available as preprints.

1. P. CHARTIER, F. MÉHATS, M. THALHAMMER, Y. ZHANG. Convergence analysis of multi-revolution compositiontime-splitting pseudo-spectral methods for linear Schrödinger equations.

Lecture note. Time-splitting spectral methods for nonlinear Schrödinger equations.

Mechthild Thalhammer (Universität Innsbruck, Austria) Operator splitting methods for nonlinear Schrödinger equations