Numerical aspects of the nonlinear Schr¨odinger equation

Numerical aspects of the nonlinear Schr¨ odinger equation

by

Christophe Besse

Laboratoire Paul Painlev´e, Universit´e Lille 1, CNRS, Inria Simpaf Team

Lille

Outline of the talk

1

Motivation

2

Numerical methods for NLS

3

Absorbing boundary conditions for the Schr¨ odinger equations

4

Some numerical experiments.

Motivation

Nonlinear Schr¨ odinger Equation (NLS)

iε∂tψ=−ε²

2∆ψ+V(x)ψ+β|ψ|²ψ , (x, t)∈R^dx×[0;T], u(x,0) =ψ0(x), x∈R^dx.

ψ(x, t): complex-valued wave function V(x): real-valued external potential

β: interaction constant (= 0: linear,+1: repulsive interaction,−1: attractive interaction)

ε: scaled Planck constant

Numerical approximations

ε= 1: standard; 0< ε1andβ=±1→semi-classical regimes efficient, convergent and accurate numerical schemes

Application of NLS

In quantum physics

Interaction between particles with quantum effect

Bose-Einstein condensation (BEC): bosons at low temperature Superfluidity

In plasma physics

wave interaction between electron and ion In semiconductor industry

In quantum chemistry

chemical interaction based on the first principle In nonlinear optics & atom laser

In fluid mechanics (water waves)

NLS is the prototype for many dispersive systems àExamples :

Davey-Stewartson systems

( i∂tu+λ∂²_xu+∂_y²u = ν|u|²u+u∂xψ, α∂²ψ+∂²ψ = χ∂x(|u|²).

Properties of NLS

Time reversible

Time transverse invariant (gauge invariant)

V(x)→V(x) +α⇒ψ→ψe^−itα/ε⇒ |ψ|unchanged Mass (wave energy) conservation

Nψ(t) :=

Z

R^d

|ψ(x, t)|²dx≡Nψ(0) :=

Z

R^d

|ψ(x,0)|²dx:=

Z

R^d

|ψ0(x)|²dx, t≥0

Energy ( or Hamiltonian) conservation

Eψ(t) :=

Z

R^d

ε²

2|∇ψ(x, t)|²+V(x)|ψ(x, t)|²+β

2|ψ(x, t)|⁴ dx≡Eψ(0), t≥0 Dispersion relation without external potential

ψ(x, t) =aei(k·x−ωt/ε) (plane wave solution) ⇒ω=ε²

|k|²+β|a|²

Numerical difficulties

Explicit vs Implicit (or computational cost) Spatial/temporal accuracy

Stability / Convergence

Keep the properties of NLS in the discretized level Time reversible & time transverse invariant Mass & energy conservation

Dispersion conservation

Resolution in the semiclassical regime: 0< ε1 Take BC into account

Outline of the talk

1

Motivation

2

Numerical methods for NLS

3

Absorbing boundary conditions for the Schr¨ odinger equations

4

Some numerical experiments.

Numerical methods for NLS

Classical schemes :

Crank-Nicolson type : Delfour-Fortin-Payre, D´uran-Sanz Serna, ...

Runge-Kutta type : Akrivis-Dougalis-Karakashian, ...

etc ...

àHigh order schemes, convergent, preserving of invariants,BUTsemi-implicit : need of a nonlinear step (Newton)

Example : Crank-Nicolson scheme (CNFD)

iεψⁿ⁺¹−ψⁿ

δt =

−ε²

2∆ +β|ψⁿ⁺¹|²+|ψⁿ|²

2 +V(·, t^n+1/2)

ψⁿ⁺¹+ψⁿ 2 = 0, ψ⁰(x) =ψ0(x), x∈R^d.

whereuⁿis the approximate value ofuat timetn=nδt.

Example : D´ uran-Sanz Serna scheme (SSFD)

iεψⁿ⁺¹−ψⁿ

δt = −ε² 2∆ +β

ψⁿ⁺¹+ψⁿ 2

2

+V(·, t^n+1/2)

!ψⁿ⁺¹+ψⁿ 2 = 0,

Possible cures : to decouple linear and nonlinear parts of (NLS)

Numerical methods for NLS

Classical schemes :

Crank-Nicolson type : Delfour-Fortin-Payre, D´uran-Sanz Serna, ...

Runge-Kutta type : Akrivis-Dougalis-Karakashian, ...

etc ...

àHigh order schemes, convergent, preserving of invariants,BUTsemi-implicit : need of a nonlinear step (Newton)

Example : Crank-Nicolson scheme (CNFD)

iεψⁿ⁺¹−ψⁿ

δt =

−ε²

2∆ +β|ψⁿ⁺¹|²+|ψⁿ|²

2 +V(·, t^n+1/2)

ψⁿ⁺¹+ψⁿ 2 = 0, ψ⁰(x) =ψ0(x), x∈R^d.

whereuⁿis the approximate value ofuat timetn=nδt.

Example : D´ uran-Sanz Serna scheme (SSFD)

ψⁿ⁺¹−ψⁿ ε²

ψⁿ⁺¹+ψⁿ2 !

ψⁿ⁺¹+ψⁿ

Numerical methods for NLS

Relaxation scheme

1 Relaxation scheme: to regard (NLS) as a Schr¨odinger-Poisson system where the Poisson equation for the potentialΥis replaced by the explicit formulaΥ =|ψ|², that is,

( Υ =|ψ|², (x, t)∈R^d×R^+∗,

iε∂tψ=−^ε2²∆ψ+V(x)ψ+βψΥ, (x, t)∈R^d×R^+∗,

Relaxation scheme (ReFD) CB ’04











Υ^n+1/2+ Υ^n−1/2

2 =|ψⁿ|², x∈R^d, iεψⁿ⁺¹−ψⁿ

δt =

−ε²

2∆ +V(x) +βΥ^n+1/2

ψⁿ⁺¹+ψⁿ

2 = 0, x∈R^d,

Advantages convergent scheme, efficient, preserves the invariants, easily adaptable Drawbacks Υ6=|u|² at the discrete level butΥ(x, t) =Rt

0∂s|u|²(x, s)ds

;make proof of convergence harder, second order, Υⁿ>0∀n?.

Numerical methods for NLS

Relaxation scheme

1 Relaxation scheme: to regard (NLS) as a Schr¨odinger-Poisson system where the Poisson equation for the potentialΥis replaced by the explicit formulaΥ =|ψ|², that is,

( Υ =|ψ|², (x, t)∈R^d×R^+∗,

iε∂tψ=−^ε2²∆ψ+V(x)ψ+βψΥ, (x, t)∈R^d×R^+∗,

Relaxation scheme (ReFD) CB ’04











Υ^n+1/2+ Υ^n−1/2

2 =|ψⁿ|², x∈R^d, iεψⁿ⁺¹−ψⁿ

δt =

−ε²

2∆ +V(x) +βΥ^n+1/2

ψⁿ⁺¹+ψⁿ

2 = 0, x∈R^d,

Advantages convergent scheme, efficient, preserves the invariants, easily adaptable Drawbacks Υ6=|u|² at the discrete level butΥ(x, t) =Rt

0∂s|u|²(x, s)ds

;make proof of convergence harder, second order, Υⁿ>0∀n?.

Numerical methods for NLS

Relaxation scheme

1 Relaxation scheme: to regard (NLS) as a Schr¨odinger-Poisson system where the Poisson equation for the potentialΥis replaced by the explicit formulaΥ =|ψ|², that is,

( Υ =|ψ|², (x, t)∈R^d×R^+∗,

iε∂tψ=−^ε2²∆ψ+V(x)ψ+βψΥ, (x, t)∈R^d×R^+∗,

Relaxation scheme (ReFD) CB ’04











Υ^n+1/2+ Υ^n−1/2

2 =|ψⁿ|², x∈R^d, iεψⁿ⁺¹−ψⁿ

δt =

−ε²

2∆ +V(x) +βΥ^n+1/2

ψⁿ⁺¹+ψⁿ

2 = 0, x∈R^d,

Advantages convergent scheme, efficient, preserves the invariants, easily adaptable Drawbacks Υ6=|u|² at the discrete level butΥ(x, t) =Rt

0∂s|u|²(x, s)ds

;make proof of convergence harder, second order, Υⁿ>0∀n?.

Numerical methods for NLS

Splitting scheme

2 Splitting scheme : Splitting methods are based on a decomposition of the flow of (NLS).

We introduceS^tthe flow of(N LS):ψ(t,·) =S^tψ0 and the two flowsX^tandY^t solutions to

∂tv=i^ε₂∆v :v(·, y) =X^tv(·,0) iε∂tw=V(x)w+β|w|²w= 0 :w(·, y) =Y^tw(·,0)

Solve nonlinear ODE analytically since∂t(|w(x, t)|²) = 0⇒ |w(x, t)|=|w(x,0)|. w(x, t) =e^−it[V(x)+β|w(x,0)|²]/εw(x,0).

Idea :to approximate the flowS^tby combining the two flowsX^tandY^t Order 1 (Lie) Z_L^t =X^tY^torZ_L^t =Y^tX^t

Order 2 (Strang) Z_S^t =X^t/2Y^tX^t/2 orZ_S^t =Y^t/2X^tY^t/2 Higher order See Descombes, Thalhammer

Numerical methods for NLS

Splitting scheme

Convergence :

Theorem (CB, Bid´ egary, Descombes ’02)

∀u0∈H²,T >0,∃Candh0such that∀h∈(0, h0],∀nsuch thatnh≤T

Z_L^hn

u0−S^nhu0

≤Chku0kH². Moreover, ifu0∈H⁴, then

Z_S^hn

u0−S^nhu0

≤Ch²ku0kH⁴.

Advantages convergent scheme, efficient, easily adaptable, very well adapted in semi-classical regimes

Drawbacks energy is not preserved, periodic boundary conditions if Fourier for the space variable.

Numerical methods for NLS

Splitting scheme

Convergence :

Theorem (CB, Bid´ egary, Descombes ’02)

∀u0∈H²,T >0,∃Candh0such that∀h∈(0, h0],∀nsuch thatnh≤T

Z_L^hn

u0−S^nhu0

≤Chku0kH². Moreover, ifu0∈H⁴, then

Z_S^hn

u0−S^nhu0

≤Ch²ku0kH⁴.

Advantages convergent scheme, efficient, easily adaptable, very well adapted in semi-classical regimes

Drawbacks energy is not preserved, periodic boundary conditions if Fourier for the space variable.

Numerical methods for NLS

Method TSSP CNFD SSFD ReFD

Time Resersible 4 4 4 4

Inv. by Gauge Change 4 6 6 6

Mass Conservation 4 4 4 4

Energy Conservation 6 4 4 4

Dispersion conservation 4 6 6 6

UnconditionalL² Stability 4 4 4 4

Convergence 4 4 4 4

Explicit Scheme 4 6 6 4

Time accuracy 2^thor4^th 2^th 2^th 2^th

Outline of the talk

1

Motivation

2

Numerical methods for NLS

3

Absorbing boundary conditions for the Schr¨ odinger equations

4

Some numerical experiments.

ABCs for Schr¨ odinger equations

NLS in R

(S)











i∂tψ+ ∆ψ+V(x, t, ψ)ψ= 0, (x, t)∈R^d×[0;T] lim

|x|→+∞ψ(x, t) = 0, t∈[0;T] ψ(x,0) =ψ0(x), x∈R^d

whereV =V(x) +f(ψ).

Motivation : the numerical resolution of Schr¨odinger equations leads to introduce a finite computational domain Ω.

=⇒introduction of a regular fictive boundaryΓ

Natural questions :

1 which are the satisfactory conditions to set onΓ×[0, T]such that the approximate solution coincides with the restriction toΩof the true solution?

2 If these conditions exist, how do we handle them numerically?

Does the energy of the approximate solution decay inΩ(stability of the numerical

ABCs for Schr¨ odinger equations

NLS in R

(S)











i∂tψ+ ∆ψ+V(x, t, ψ)ψ= 0, (x, t)∈R^d×[0;T] lim

|x|→+∞ψ(x, t) = 0, t∈[0;T] ψ(x,0) =ψ0(x), x∈R^d

whereV =V(x) +f(ψ).

Motivation : the numerical resolution of Schr¨odinger equations leads to introduce a finite computational domain Ω.

=⇒introduction of a regular fictive boundaryΓ

Natural questions :

1 which are the satisfactory conditions to set onΓ×[0, T]such that the approximate solution coincides with the restriction toΩof the true solution?

If these conditions exist, how do we handle them numerically?

Domain truncation

Problem : Mesh an unbounded domain (here in 1D)R^d×[0;T] t

0 x

ψ

TruncationR×[0;T]−→ΩT:= ]xl, xr[×[0;T]

Introduction of a fictitious boundaryΣT :=Σ×[0;T]withΣ :=∂Ω ={xl, xr} ⇒ BC on the boundaryΣT

The boundary condition onx_lmust represent the effect of the potential on]− ∞, x_l]. Expression of the boundary condtion with the help of the Dirichlet-to-Neumann map:

∂nψ+iΛ⁺ψ= 0, onΣT

Domain truncation

Problem : Mesh an unbounded domain (here in 1D)R^d×[0;T]

ΩT

T t

xl 0 xr

ΣT

x ψ

TruncationR×[0;T]−→ΩT:= ]xl, xr[×[0;T]

Introduction of a fictitious boundaryΣT:=Σ×[0;T]withΣ :=∂Ω ={xl, xr} ⇒ BC on the boundaryΣT

The boundary condition onx_lmust represent the effect of the potential on]− ∞, x_l]. Expression of the boundary condtion with the help of the Dirichlet-to-Neumann map:

∂nψ+iΛ⁺ψ= 0, onΣT

Domain truncation

Problem : Mesh an unbounded domain (here in 1D)R^d×[0;T]

ΩT

T t

xl 0 xr

ΣT

x f(ψ, ∂nψ) = 0 ψ

TruncationR×[0;T]−→ΩT:= ]xl, xr[×[0;T]

Introduction of a fictitious boundaryΣT:=Σ×[0;T]withΣ :=∂Ω ={xl, xr} ⇒ BC on the boundaryΣT

The boundary condition onx_lmust represent the effect of the potential on]− ∞, x_l].

Expression of the boundary condtion with the help of the Dirichlet-to-Neumann map:

∂nψ+iΛ⁺ψ= 0, onΣT

What happen if we do not take BC into account

PotentialV(x) =x Initial datum:

gaussian function ψ0(x) =e^−x2+10ix

Domain:Ω = [−5; 15]

−5 0 5 10 15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

| u(x,t) |

Evolution of|ψ|w.r.t time

∆t= 0.2

Homogeneous Dirichlet Boundary Conditions:

ψ_|Σ= 0

⇒Parasistic reflexion

t

Exact Solution

t

0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.3 0.4 0.5 0.6 0.7 0.8 0.9

What happen if we do not take BC into account

PotentialV(x) =x Initial datum:

gaussian function ψ0(x) =e^−x2+10ix

Domain:Ω = [−5; 15]

−5 0 5 10 15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

| u(x,t) |

Evolution of|ψ|w.r.t time

∆t= 0.2

Homogeneous Dirichlet Boundary Conditions:

ψ_|Σ= 0

⇒Parasistic reflexion

t

Approximated numerical solution

t

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

The 1D Eq. without potential

1 Splitting between interior and exterior problems Interior Problem







(i∂_t+∂²_x)v= 0, x∈Ω, t >0,

∂xv=∂xw, x∈Σ, t >0, v(x,0) =ψ₀(x), x∈Ω.

Exterior problem















(i∂_t+∂_x²)w= 0, x∈Ω, t >0, w(x, t) =v(x, t), x=x_l,r, t >0,

lim

|x|→+∞w(x, t) = 0, t >0,

w(x,0) =0, x∈Ω.

000 111 000 111

problem interior problem

left exterior problem

right exterior

(x,t)

x_L x_R

output: input:

(x ,t)L x(x ,t)L

Dirichlet data Neumann data

v

v w

2 Laplace transform w.r.t time (t→τ) onΩr: ODE inx

3 Argument: w∈L²(Ωr)to select the outgoing wave

4 Inverse Laplace transform

5 Exact boundary condition onxr: ∂xv(x, t)|^x=xr=−e^−iπ/4∂^1/2_t v(xr, t) where ∂^1/2f(x, t) = 1

√ ∂

Ztf(x, s)|^x=xr

√ ds fractional derivative op-

The 1D Eq. without potential

The problem(S)is thus transformed in(Sapp)

The Schr¨ odinger Eq. in Ω

(Sapp)











i∂tψ+ ∆ψ= 0, (x, t)∈ΩT

∂nψ+e^−iπ/4∂_t^1/2ψ= 0, onΣT, ψ(x,0) =ψ0(x), x∈ΩT

Remark:

∂_x²+i∂t= (∂n+i√

i∂t)(∂n−i√ i∂t)

Extension to simple cases (1)

V = 0

Expression of the exact Dirichlet-to-Neumann operator

∂nψ+e^−iπ/4∂_t^1/2ψ= 0, onΣT. (ABC0)

V = V

constant at the exterior of Ω

∂nψ+e^−iπ/4e^itV`,r∂_t^1/2

e^−itV`,rψ

= 0, onΣT.

V = V (t) : Gauge change

We set: v(x, t) =ψ(x, t)e^−iV(t) with V(t) = Z t

0

V(s)ds.

Then i∂tψ= (i∂tv−V(t)v)e^iV(t), wherevis solution to the equation without potential

Non constant potentials

IfV =V(x, t), then the Laplace transform can not be used anymore.

One have to introduce a new tool:pseudodifferential calculus.

Pseudodifferential Operators in 1D

A pseudodifferential operatorP(x, t, ∂t)is described by its total symbolp(x, t, τ)in the Fourier space (τis the covariable oft)

P(x, t, ∂t)u(x, t) =Ft⁻¹ p(x, t, τ)ˆu(x, τ)

= Z

R

p(x, t, τ)u(x, τˆ )e^itτdτ

Notations: P=Op(p) , p(x, t, τ) =σ(P(x, t, ∂t))

Non constant potentials

IfV =V(x, t), then the Laplace transform can not be used anymore.

One have to introduce a new tool:pseudodifferential calculus.

Pseudodifferential Operators in 1D

A pseudodifferential operatorP(x, t, ∂t)is described by its total symbolp(x, t, τ)in the Fourier space (τis the covariable oft)

P(x, t, ∂t)u(x, t) =Ft⁻¹ p(x, t, τ)u(x, τˆ )

= Z

R

p(x, t, τ)u(x, τˆ )e^itτdτ

Notations: P=Op(p) , p(x, t, τ) =σ(P(x, t, ∂t))

Non constant potentials

Examples

The fractional operators ∂

and I

∂^1/2_t f(t) = 1

√π∂t

Z t 0

f(s)

√t−sds

I_t^α/2f(t) = 1 Γ(α/2)

Zt 0

(t−s)^α/2−1f(s)ds

Nonlocal w.r.t time convolution operator

Operator ∂t ∂_t^1/2 I_t^1/2 It

↓ ↓ ↓ ↓

Symbol iτ e^−iπ/4√

−τ e^iπ/4

√−τ

1 iτ

Non constant potentials

CaseV = 0

TBC: ∂nψ+e^−iπ/4∂_t^1/2ψ= 0, onΣ_T.

∂nψ−iOp √

−τ

ψ= 0, onΣT.

Case constantV =V

TBC: ∂nψ+e^−iπ/4e^itV∂_t^1/2 e^−itVψ

= 0, onΣ_T.

∂nψ−iOp√

−τ +V

(ψ) = 0, onΣT.

Lemma

Ifais a symbol belonging toS^mindependent oft, andV =V(x), then Op(a(τ −V(x)))ψ=e^itV(x)Op(a(τ))

e^−itV(x)ψ

CaseV =V(t): Gauge change Antoine, Besse et Descombes, 2006

∂nψ−i e^iV(t)Op √

−τ e^−iV(t)ψ

= 0, onΣT.

Non constant potentials

CaseV = 0

TBC: ∂nψ+e^−iπ/4∂_t^1/2ψ= 0, onΣ_T.

∂nψ−iOp √

−τ

ψ= 0, onΣT.

Case constantV =V TBC: ∂nψ−ie^itVOp √

−τ

e^−itVψ

= 0, onΣ_T.

∂nψ−iOp√

−τ +V

(ψ) = 0, onΣT.

Lemma

Ifais a symbol belonging toS^mindependent oft, andV =V(x), then Op(a(τ −V(x)))ψ=e^itV(x)Op(a(τ))

e^−itV(x)ψ

CaseV =V(t): Gauge change Antoine, Besse et Descombes, 2006

∂nψ−i e^iV(t)Op √

−τ e^−iV(t)ψ

= 0, onΣT.

Non constant potentials

CaseV = 0

TBC: ∂nψ+e^−iπ/4∂_t^1/2ψ= 0, onΣ_T.

∂nψ−iOp √

−τ

ψ= 0, onΣT.

Case constantV =V TBC: ∂nψ−ie^itVOp √

−τ

e^−itVψ

= 0, onΣ_T.

∂nψ−iOp√

−τ +V

(ψ) = 0, onΣT.

Lemma

Ifais a symbol belonging toS^mindependent oft, andV =V(x), then Op(a(τ −V(x)))ψ=e^itV(x)Op(a(τ))

e^−itV(x)ψ

CaseV =V(t): Gauge change Antoine, Besse et Descombes, 2006

∂nψ−i e^iV(t)Op √

−τ e^−iV(t)ψ

= 0, onΣT.

Non constant potentials

CaseV = 0

TBC: ∂nψ+e^−iπ/4∂_t^1/2ψ= 0, onΣ_T.

∂nψ−iOp √

−τ

ψ= 0, onΣT.

Case constantV =V TBC: ∂nψ−ie^itVOp √

−τ

e^−itVψ

= 0, onΣ_T.

∂nψ−iOp√

−τ +V

(ψ) = 0, onΣT.

Lemma

Ifais a symbol belonging toS^mindependent oft, andV =V(x), then Op(a(τ −V(x)))ψ=e^itV(x)Op(a(τ))

e^−itV(x)ψ

CaseV =V(t): Gauge change Antoine, Besse et Descombes, 2006

With potential V = V (x, t) + f (|ψ|) : two strategies

Use of pseudodifferential calculus

No More Exact! : Artifical Boundary Condition

1) Gauge change

v(x, t) =e^−iV(x,t)ψ(x, t), with V(x, t) = Z t

0 V(x, s, ψ)ds.

Involve operators

e^iV(x,t)Op √

−τ

e^−iV(x,t)ψ

2) Direct method

No Gauge change Involve operators

Opp

−τ +V(x, t, ψ) (ψ)

Equivalent strategies forV =V(x), non equivalent forV =V(x, t)

With potential V = V (x, t) + f (|ψ|) : two strategies

Nonlinearities and general repulsive potentialsx∂xV >0forx∈Ω

V = f (x, ψ)

V =f(x, ψ)andV(x, t) =Rt

0f(x, ψ(x, s))ds Absorbing boundary conditions (ABC) forM= 4:

ABC⁴₁: ∂nψ+e^−iπ/4e^iV∂^1/2_t

e^−iVψ

−i∂nV 4 e^iVIt

e^−iVψ

= 0 ABC⁴₂: ∂nψ−ip

i∂t+Vψ+1

4∂nV(i∂t+V)⁻¹ψ= 0

Extension to higher dimensions

Take into account the geometry: convex set with general boundary, smooth, with curvatureκ.

n Ω

Generalized coordinates systemof the bound- ary with respect to normal variablerand curvilinear abscissas

∆ =∂_r²+κr∂r+h⁻¹∂s h⁻¹∂s

κr=h⁻¹κ: curvature of a parallel surfaceΣr toΣ h(r, s) = 1 +rκ

⇒ L=∂_r²+κr∂r+i∂t+h⁻¹∂s h⁻¹∂s

+V Schr¨odinger equation with variable coefficients:

pseudo-differential calculus

τ

s n

r M(s)

M^′(r,s)

Nonlocality

Nonlocal both in space and time.

Localizing in space: two approaches, valid for both strategies

”Taylor” approach: Taylor expansion of the symbols for|τ| ξ²

−τ−ξ²+V =−τ

1 +ξ² τ −V

τ

Thereby:

p−τ−ξ²+V ≈√

−τ

1 + ξ² 2τ − V

2τ

=√

−τ−ξ² 2

√1

−τ +V 2

√1

−τ

=⇒Localizing in space only

Pad´e approximation approach:

Opp

−τ−ξ²+V

∼p

i∂t+ ∆Σ+V modOP S⁻¹ formal approximation of√

·by Pad´e approximants=⇒Localizing both in spaceAND

Conclusion

Two possible approaches for each strategy, so 4 families of ABC.

”Taylor” approach ”Pad´e” approach

Gauge change ABC_1,T^M ABC_1,P^M

Direct method ABC_2,T^M ABC_2,P^M

↓ ↓

∂_t^1/2,I^1/2_t ,It Opp

−τ−ξ² or v=e^−iVu Opp

−τ−ξ²+V

ABC: Taylor approach

Gauge change

ABC²_1,T ∂nψ+e^−iπ/4e^iV∂_t^1/2 e^−iVψ

+κ 2ψ

ABC³_1,T −e^iπ/4e^iV κ²

8 +∆Σ

2 +i∂sV∂s+1

2(i∂²_sV −(∂sV)²)

I^1/2_t e^−iVψ

ABC⁴_1,T +ie^iV

∂s(κ∂s)

2 +κ³+∂²_sκ

8 +i∂sκ∂sV 2

It

e^−iVψ

−isg(∂nV) 4

p|∂nV|e^iVIt

p

|∂nV|e^−iVψ

= 0

Direct method

ABC²_2,T ∂nψ+e^−iπ/4∂_t^1/2ψ+κ 2ψ

ABC³_2,T −e^iπ/4 κ²

8 +∆Σ

2

I_t^1/2ψ−e^iπ/4sg(V) 2

p|V|I_t^1/2p

|V|ψ ABC⁴_2,T +i

∂s(κ∂s)

2 +κ³+∂_s²κ 8

Itψ−isg(∂nV) 4

p|∂nV|It

p|∂nV|ψ

= 0

ABC: Pad´ e approach

Gauge change

ABC¹_1,P ∂nψ−ie^iVp i∂t+ ∆Σ

e^−iVψ

ABC²_1,P + κ

2ψ+∂sVe^iV∂s(i∂t+ ∆Σ)^−1/2 e^−iVψ

− κ

2e^iV(i∂t+ ∆Σ)⁻¹∆Σ

e^−iVψ

= 0

Direct method

ABC¹_2,P ∂nψ−ip

i∂t+ ∆Σ+V ψ ABC²_2,P + κ

2ψ−κ

2(i∂t+ ∆Σ+V)⁻¹∆Σψ= 0

Taylor approach conditions

Approximations of∂_t^1/2,I_t^1/2,It by discrete convolutions, linked to the Crank-Nicolson scheme⇒trapezoidal rule[Schmidt - Yevick (97), Antoine - Besse (03)]

∂_t^1/2f(tⁿ)≈ r 2

∆t

n

X

k=0

βn−kf^k

I_t^1/2f(tⁿ)≈ r∆t

2

n

X

k=0

αn−kf^k

Itf(tⁿ)≈∆t 2

n

X

k=0

γn−kf^k











(α0, α1, α2, . . .) =

1,1,1 2,1

2,3 8,3

8, . . .

β_k= (−1)^kα_k,∀k≥0 (γ0, γ1, γ2, . . .) = (1,2,2,2, . . .)

Pad´ e approach conditions

[Bruneau - Di Menza (95), Szeftel (04)]

Approximation of√

i∂t+ ∆Σ+V

Rational approximation of the square root by Pad´e approximants

√z≈Rm(z) =

m

X

k=0

a^m_k −

m

X

k=1

a^m_kd^m_k z+d^m_k

In theABC_2,P^M conditions:

pi∂t+ ∆Σ+V ; Rm(i∂t+ ∆Σ+V) Rm

⇒ p

i∂t+ ∆Σ+V ψ ≈

m

X

k=0

a^m_k

! ψ−

m

X

k=1

a^m_kd^m_k(i∂t+ ∆Σ+V +d^m_k)⁻¹ψ

| {z }

ϕ_k

Outline of the talk

1

Motivation

2

Numerical methods for NLS

3

Absorbing boundary conditions for the Schr¨ odinger equations

4

Some numerical experiments.

Numerical application

1d linear caseu(x, t) = s

i

−4t+iexp

−ix²−k0x+k₀²t

−4t+i

k0= 8, Ω= [−5,5], N= 1024,∆t= 10⁻³

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

|u|

Evolution of |u|

t=0 t=0.15 t=0.30 t=0.45

2D linear case Explicit solution (2D)

u(x1, x2, t) = i i−4texp

−ix²₁+x²₂+ 5ix1+ 25it i−4t

.

Finite Element Approximation(P¹):Ωi=D(0,10),3278triangles,∆t= 10⁻².

2D linear and nonlinear case with potentials

Linear, Circular domain,

V(x, y) = 5(x²+y²) Linear, Mediator shaped domain, V(x, y) = 5p

x²+y²

Nonlinear, Circular domain, V(x, y) = 0

Nonlinear, Circular domain, V(x, y) =x²+y²