Numerical aspects of the nonlinear Schro dinger equation. Christophe Besse

Transcription

1 Numerical aspects of the nonlinear Schro dinger equation by Christophe Besse Laboratoire Paul Painleve, Universite Lille 1, CNRS, Inria Simpaf Team Lille

2 Outline of the talk 1 Motivation 2 Numerical methods for NLS 3 Absorbing boundary conditions for the Schrödinger equations 4 Some numerical experiments.

3 Motivation Nonlinear Schrödinger Equation (NLS) iε tψ = ε2 2 ψ + V (x)ψ + β ψ 2 ψ, (x, t) R d x [0; T ], u(x, 0) = ψ 0 (x), x R d x. 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 < ε 1 and β = ±1 semi-classical regimes efficient, convergent and accurate numerical schemes compute efficient boundary conditions to deal with bounded computational domains.

4 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 Examples : Davey-Stewartson systems (water waves) { i tu + λ 2 x u + 2 y u = ν u 2 u + u xψ, α 2 x ψ + 2 y ψ = χ x( u 2 ).

5 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 N ψ (t) := ψ(x, t) 2 dx N ψ (0) := R d ψ(x, 0) 2 dx := R d ψ 0 (x) 2 dx, R d t 0 Energy ( or Hamiltonian) conservation ε 2 E ψ (t) := R d 2 ψ(x, t) 2 + V (x) ψ(x, t) 2 + β 2 ψ(x, t) 4 dx E ψ (0), t 0 Dispersion relation without external potential ψ(x, t) = ae i(k x ωt/ε) (plane wave solution) ω = ε2 2 k 2 + β a 2

6 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 < ε 1 Take BC into account

7 Outline of the talk 1 Motivation 2 Numerical methods for NLS 3 Absorbing boundary conditions for the Schrödinger equations 4 Some numerical experiments.

8 Numerical methods for NLS Classical schemes : Crank-Nicolson type : Delfour-Fortin-Payre, Dúran-Sanz Serna,... Runge-Kutta type : Akrivis-Dougalis-Karakashian,... etc... High order schemes, convergent, preserving of invariants, BUT semi-implicit : need of a nonlinear step (Newton) Example : Crank-Nicolson scheme (CNFD) iε ψn+1 ψ n = ( ε22 δt + β ψn ψ n 2 ) ψ + V (, t n+1/2 n+1 + ψ n ) 2 2 ψ 0 (x) = ψ 0 (x), x R d. = 0, where u n is the approximate value of u at time t n = nδt. Example : Dúran-Sanz Serna scheme (SSFD) ( iε ψn+1 ψ n = ε2 δt 2 + β ψ n+1 + ψ n ) V (, t n+1/2 ψ n+1 + ψ n ) 2 ψ 0 (x) = ψ 0 (x), x R d. = 0,

9 Numerical methods for NLS Classical schemes : Crank-Nicolson type : Delfour-Fortin-Payre, Dúran-Sanz Serna,... Runge-Kutta type : Akrivis-Dougalis-Karakashian,... etc... High order schemes, convergent, preserving of invariants, BUT semi-implicit : need of a nonlinear step (Newton) Example : Crank-Nicolson scheme (CNFD) iε ψn+1 ψ n = ( ε22 δt + β ψn ψ n 2 ) ψ + V (, t n+1/2 n+1 + ψ n ) 2 2 ψ 0 (x) = ψ 0 (x), x R d. = 0, where u n is the approximate value of u at time t n = nδt. Example : Dúran-Sanz Serna scheme (SSFD) ( iε ψn+1 ψ n = ε2 δt 2 + β ψ n+1 + ψ n ) V (, t n+1/2 ψ n+1 + ψ n ) 2 ψ 0 (x) = ψ 0 (x), x R d. = 0, Possible cures : to decouple linear and nonlinear parts of (NLS)

10 Numerical methods for NLS Relaxation scheme 1 Relaxation scheme: to regard (NLS) as a Schrödinger-Poisson system where the Poisson equation for the potential Υ is replaced by the explicit formula Υ = ψ 2, that is, { Υ = ψ 2, (x, t) R d R +, iε tψ = ε2 2 ψ + V (x)ψ + βψυ, (x, t) Rd R +,

11 Numerical methods for NLS Relaxation scheme 1 Relaxation scheme: to regard (NLS) as a Schrödinger-Poisson system where the Poisson equation for the potential Υ is replaced by the explicit formula Υ = ψ 2, that is, { Υ = ψ 2, (x, t) R d R +, iε tψ = ε2 2 ψ + V (x)ψ + βψυ, (x, t) Rd R +, Relaxation scheme (ReFD) CB 04 Υ n+1/2 + Υ n 1/2 = ψ n 2, x R d, 2 iε ψn+1 ψ n δt = ( ε22 ) ψ n+1 + V (x) + + ψ n βυn+1/2 = 0, x R d, 2

12 Numerical methods for NLS Relaxation scheme 1 Relaxation scheme: to regard (NLS) as a Schrödinger-Poisson system where the Poisson equation for the potential Υ is replaced by the explicit formula Υ = ψ 2, that is, { Υ = ψ 2, (x, t) R d R +, iε tψ = ε2 2 ψ + V (x)ψ + βψυ, (x, t) Rd R +, Relaxation scheme (ReFD) CB 04 Υ n+1/2 + Υ n 1/2 = ψ n 2, x R d, 2 iε ψn+1 ψ n δt = ( ε22 ) ψ n+1 + V (x) + + ψ n βυn+1/2 = 0, x R d, 2 Advantages convergent scheme, efficient, preserves the invariants, easily adaptable Drawbacks Υ u 2 at the discrete level but Υ(x, t) = t 0 s u 2 (x, s)ds make proof of convergence harder, second order, Υ n > 0 n?.

13 Numerical methods for NLS Splitting scheme 2 Splitting scheme : Splitting methods are based on a decomposition of the flow of (NLS). We introduce S t the flow of (NLS) : ψ(t, ) = S t ψ 0 and the two flows X t and Y t solutions to tv = i ε 2 v : v(, y) = Xt v(, 0) iε tw = V (x)w + β w 2 w = 0 : w(, y) = Y t w(, 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 ]/ε w(x, 0). Idea : to approximate the flow S t by combining the two flows X t and Y t Order 1 (Lie) Z t L = Xt Y t or Z t L = Y t X t Order 2 (Strang) Z t S = Xt/2 Y t X t/2 or Z t S = Y t/2 X t Y t/2 Higher order See Descombes, Thalhammer

14 Numerical methods for NLS Splitting scheme Convergence : Theorem (CB, Bidégary, Descombes 02) u 0 H 2, T > 0, C and h 0 such that h (0, h 0 ], n such that nh T ( ) n ZL h u0 S nh u 0 Ch u0 H 2. ( ) Moreover, if u 0 H 4 n, then ZS h u0 S nh u 0 Ch 2 u 0 H 4.

15 Numerical methods for NLS Splitting scheme Convergence : Theorem (CB, Bidégary, Descombes 02) u 0 H 2, T > 0, C and h 0 such that h (0, h 0 ], n such that nh T ( ) n ZL h u0 S nh u 0 Ch u0 H 2. ( ) Moreover, if u 0 H 4 n, then ZS h u0 S nh u 0 Ch 2 u 0 H 4. 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.

16 Numerical methods for NLS Method TSSP CNFD SSFD ReFD Time Resersible Inv. by Gauge Change Mass Conservation Energy Conservation Dispersion conservation Unconditional L 2 Stability Convergence Explicit Scheme Time accuracy 2 th or 4 th 2 th 2 th 2 th

17 Outline of the talk 1 Motivation 2 Numerical methods for NLS 3 Absorbing boundary conditions for the Schrödinger equations 4 Some numerical experiments.

18 ABCs for Schrödinger equations NLS in R d i tψ + ψ + V (x, t, ψ) ψ = 0, (x, t) R d [0; T ] (S) lim ψ(x, t) = 0, t [0; T ] x + ψ(x, 0) = ψ 0 (x), x R d where V = V (x) + f(ψ). Motivation : the numerical resolution of Schrödinger equations leads to introduce a finite computational domain Ω. = introduction of a regular fictive boundary Γ Natural questions : 1 which are the satisfactory conditions to set on Γ [0, T ] such that the approximate solution coincides with the restriction to Ω of the true solution? 2 If these conditions exist, how do we handle them numerically? 3 Does the energy of the approximate solution decay in Ω (stability of the numerical schemes)?

19 ABCs for Schrödinger equations NLS in R d i tψ + ψ + V (x, t, ψ) ψ = 0, (x, t) R d [0; T ] (S) lim ψ(x, t) = 0, t [0; T ] x + ψ(x, 0) = ψ 0 (x), x R d where V = V (x) + f(ψ). Motivation : the numerical resolution of Schrödinger equations leads to introduce a finite computational domain Ω. = introduction of a regular fictive boundary Γ Natural questions : 1 which are the satisfactory conditions to set on Γ [0, T ] such that the approximate solution coincides with the restriction to Ω of the true solution? 2 If these conditions exist, how do we handle them numerically? 3 Does the energy of the approximate solution decay in Ω (stability of the numerical schemes)?

20 Domain truncation Problem : Mesh an unbounded domain (here in 1D) R d [0; T ] t ψ 0 x

21 Domain truncation Problem : Mesh an unbounded domain (here in 1D) R d [0; T ] t T Σ T Ω T ψ x l 0 x r x Truncation R [0; T ] Ω T := ]x l, x r[ [0; T ] Introduction of a fictitious boundary Σ T := Σ [0; T ] with Σ := Ω = {x l, x r} BC on the boundaryσ T

22 Domain truncation Problem : Mesh an unbounded domain (here in 1D) R d [0; T ] t T Σ T Ω T f(ψ, n ψ) = 0 ψ x l 0 x r x Truncation R [0; T ] Ω T := ]x l, x r[ [0; T ] Introduction of a fictitious boundary Σ T := Σ [0; T ] with Σ := Ω = {x l, x r} BC on the boundaryσ T The boundary condition on x l must 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, on Σ T

23 What happen if we do not take BC into account t Potential V (x) = x Initial datum: gaussian function ψ 0 (x) = e x2 +10ix Domain: Ω = [ 5; 15] u(x,t) x 2 Exact Solution Evolution of ψ w.r.t time t = t x x

24 What happen if we do not take BC into account t Potential V (x) = x Initial datum: gaussian function ψ 0 (x) = e x2 +10ix Domain: Ω = [ 5; 15] u(x,t) x Evolution of ψ w.r.t time t = 0.2 Approximated numerical solution Homogeneous Dirichlet Boundary Conditions: ψ Σ = 0 Parasistic reflexion t x x

25 The 1D Eq. without potential L L 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) = ψ 0(x), x Ω. Exterior problem (i t + 2 x )w = 0, x Ω, t > 0, w(x, t) = v(x, t), x = x l,r, t > 0, lim w(x, t) = 0, t > 0, x + w(x, 0) = 0, x Ω. left exterior problem interior problem right exterior problem output: Neumann data wx(x,t) input: Dirichlet data v(x,t) v (x,t) xl xr 2 Laplace transform w.r.t time (t τ) on Ω r: ODE in x 3 Argument: w L 2 (Ω r) to select the outgoing wave 4 Inverse Laplace transform 5 Exact boundary condition on x r: xv(x, t) x=xr = e iπ/4 1/2 t v(x r, t) TBC on Σ : where 1/2 t f(x, t) = 1 π t t 0 f(x, s) x=xr t s ds nv + e iπ/4 1/2 t v = 0, on Σ [0; T ]. fractional derivative operator of order 1/2

26 The 1D Eq. without potential The problem (S) is thus transformed in (S app) The Schrödinger Eq. in Ω i tψ + ψ = 0, (x, t) Ω T (S app) nψ + e iπ/4 1/2 t ψ = 0, on Σ T, ψ(x, 0) = ψ 0 (x), x Ω T Remark: 2 x + i t = ( n + i i t)( n i i t)

27 Extension to simple cases (1) V = 0 Expression of the exact Dirichlet-to-Neumann operator nψ + e iπ/4 1/2 t ψ = 0, on Σ T. (ABC 0 ) V = V l,r constant at the exterior of Ω ( ) nψ + e iπ/4 e itv l,r 1/2 t e itv l,r ψ = 0, on Σ T. V = V (t) : Gauge change t We set: v(x, t) = ψ(x, t)e iv(t) with V(t) = V (s) ds. 0 Then i tψ = (i tv V (t) v) e iv(t), where v is solution to the equation without potential Exact condition ABC 0 for v : nv + e iπ/4 1/2 t v = 0, on Σ T ( ) We come back to ψ : nψ + e iπ/4 e iv(t) 1/2 t e iv(t) ψ = 0, on Σ T

28 Non constant potentials If V = V (x, t), then the Laplace transform can not be used anymore. One have to introduce a new tool: pseudodifferential calculus.

29 Non constant potentials If V = 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 operator P (x, t, t) is described by its total symbol p(x, t, τ) in the Fourier space (τ is the covariable of t) P (x, t, t) u(x, t) = Ft 1 ( ) p(x, t, τ) û(x, τ) = p(x, t, τ) û(x, τ)e itτ dτ R Notations: P = Op(p), p(x, t, τ) = σ(p (x, t, t))

30 Non constant potentials Examples The fractional operators 1/2 t and I α/2 t 1/2 t f(t) = 1 t f(s) t ds π 0 t s I α/2 1 t t f(t) = (t s) α/2 1 f(s) ds Γ(α/2) 0 Nonlocal w.r.t time convolution operator Operator t 1/2 t I 1/2 t I t Symbol iτ e iπ/4 τ e iπ/4 τ 1 iτ Class OP S 1 OP S 1/2 OP S 1/2 OP S 1

31 Non constant potentials Case V = 0 TBC: nψ + e iπ/4 1/2 t ψ = 0, on Σ T. Case constant V = V TBC: nψ + e iπ/4 e itv 1/2 ( t e itv ψ ) = 0, on Σ T.

32 Non constant potentials Case V = 0 TBC: nψ + e iπ/4 1/2 t ψ = 0, on Σ T. nψ i Op ( τ ) ψ = 0, on Σ T. Case constant V = V TBC: nψ i e itv Op ( τ ) ( e itv ψ ) = 0, on Σ T.

33 Non constant potentials Case V = 0 TBC: nψ + e iπ/4 1/2 t ψ = 0, on Σ T. nψ i Op ( τ ) ψ = 0, on Σ T. Case constant V = V TBC: nψ i e itv Op ( τ ) ( e itv ψ ) = 0, on Σ T. ( τ ) nψ i Op + V (ψ) = 0, on Σ T. Lemma If a is a symbol belonging to S m independent of t, and V = V (x), then ( ) Op (a(τ V (x))) ψ = e itv (x) Op (a(τ)) e itv (x) ψ

34 Non constant potentials Case V = 0 TBC: nψ + e iπ/4 1/2 t ψ = 0, on Σ T. nψ i Op ( τ ) ψ = 0, on Σ T. Case constant V = V TBC: nψ i e itv Op ( τ ) ( e itv ψ ) = 0, on Σ T. ( τ ) nψ i Op + V (ψ) = 0, on Σ T. Lemma If a is a symbol belonging to S m independent of t, and V = V (x), then ( ) Op (a(τ V (x))) ψ = e itv (x) Op (a(τ)) e itv (x) ψ Case V = V (t) : Gauge change Antoine, Besse et Descombes, 2006 nψ i e iv(t) Op ( τ ) ( ) e iv(t) ψ = 0, on Σ T.

35 With potential V = V (x, t) + f( ψ ): two strategies Use of pseudodifferential calculus No More Exact! : Artifical Boundary Condition 1) Gauge change t v(x, t) = e iv(x,t) ψ(x, t), with V(x, t) = V (x, s, ψ) ds. 0 Involve operators e iv(x,t) Op ( τ ) ( ) e iv(x,t) ψ 2) Direct method No Gauge change Involve operators ( ) Op τ + V (x, t, ψ) (ψ) Equivalent strategies for V = V (x), non equivalent for V = V (x, t) In both cases, we still have to approximate Boundary Conditions, at different order M. Antoine, Besse et Klein, J. Comput. Phys., 2009

36 With potential V = V (x, t) + f( ψ ): two strategies Nonlinearities and general repulsive potentials x xv > 0 for x Ω V = f(x, ψ) V = f(x, ψ) and V(x, t) = t 0 f(x, ψ(x, s))ds Absorbing boundary conditions (ABC) for M = 4: ( ) ABC1 4 : nψ + e iπ/4 e iv 1/2 t e iv ψ i nv ( ) 4 eiv I t e iv ψ = 0 ABC 4 2 : nψ i i t + V ψ nv (i t + V ) 1 ψ = 0

37 Extension to higher dimensions Take into account the geometry: convex set with general boundary, smooth, with curvature κ. Ω n Generalized coordinates system of the boundary with respect to normal variable r and curvilinear abscissa s = 2 r + κ r r + h 1 s ( h 1 s ) κ r = h 1 κ : curvature of a parallel surface Σ r to Σ h(r, s) = 1 + rκ L = 2 r + κ r r + i t + h 1 s ( h 1 s ) + V τ s M (s) r n M (r, s) Schrödinger equation with variable coefficients: pseudo-differential calculus t t dual variable τ x r y s dual variable ξ Σ Σ r

38 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 ξ 2 Thereby: τ ξ 2 + V = τ (1 + ξ2 τ V ) τ τ ξ 2 + V τ (1 + ξ2 2τ V ) = τ ξ2 2τ 2 = Localizing in space only 1 τ + V 2 1 τ Padé approximation approach: Op ( τ ξ 2 + V ) i t + Σ + V mod OP S 1 formal approximation of by Padé approximants = Localizing both in space AND time

39 Conclusion Two possible approaches for each strategy, so 4 families of ABC. Taylor approach Padé approach Gauge change ABC M 1,T ABC M 1,P Direct method ABC M 2,T ABC M 2,P ( 1/2 t, I 1/2 t, I t Op τ ξ 2) ( ) v = e iv u Op τ ξ 2 + V or

40 ABC: Taylor approach Gauge change ABC 2 1,T nψ + e iπ/4 e iv 1/2 t ( ) e iv ψ + κ 2 ψ ( κ ABC 3 1,T e iπ/4 e iv Σ 2 + i sv s + 1 ) 2 (i 2 s V ( sv)2 ) ) ( ) I t e iv ψ ( s(κ s) ABC 4 1,T + ie iv + κ3 + s 2κ + i sκ sv i sg( nv ) ( nv e iv I t nv e ψ) iv = 0 4 Direct method ABC 2 2,T nψ + e iπ/4 1/2 t ψ + κ 2 ψ ( κ ABC 3 2,T e iπ/ Σ 2 ( s(κ s) ABC 4 2,T + i 2 ) I 1/2 + κ3 + 2 s κ 8 iπ/4 sg(v ) t ψ e 2 ) I tψ i sg( nv ) 4 1/2 ( ) V I t V ψ ( ) I 1/2 t e iv ψ nv I t ( nv ψ) = 0

41 ABC: Padé approach Gauge change ABC 1 1,P nψ ie ) iv i t + Σ (e iv ψ ABC 2 1,P + κ ( ) 2 ψ + sveiv s (i t + Σ ) 1/2 e iv ψ κ ) 2 eiv (i t + Σ ) 1 Σ (e iv ψ = 0 Direct method ABC 1 2,P nψ i i t + Σ + V ψ ABC 2 2,P + κ 2 ψ κ 2 (i t + Σ + V ) 1 Σ ψ = 0

42 Taylor approach conditions Approximations of 1/2 t, I 1/2 t, I t by discrete convolutions, linked to the Crank-Nicolson scheme trapezoidal rule [Schmidt - Yevick (97), Antoine - Besse (03)] 1/2 2 t f(t n ) t I 1/2 t t f(t n ) 2 I tf(t n ) t 2 n β n k f k k=0 n α n k f k k=0 n γ n k f k k=0 (α 0, α 1, α 2,...) = (1, 1, 12, 12, 38, 38 ),... β k = ( 1) k α k, k 0 (γ 0, γ 1, γ 2,...) = (1, 2, 2, 2,...)

43 Padé approach conditions Approximation of i t + Σ + V Rational approximation of the square root by Padé approximants [Bruneau - Di Menza (95), Szeftel (04)] m z Rm(z) = a m k k=0 m a m k dm k z + d m k=1 k In the ABC M 2,P conditions: i t + Σ + V R m (i t + Σ + V ) i t + Σ + V ψ R m ( m ) m a m k ψ a m k dm k (i t + Σ + V + d m k ) 1 ψ }{{} k=0 k=1 ϕ k Auxiliary functions ϕ k, solutions of a Schrödinger equation on Σ T.

44 Outline of the talk 1 Motivation 2 Numerical methods for NLS 3 Absorbing boundary conditions for the Schrödinger equations 4 Some numerical experiments.

45 Numerical application ( i ix 2 1d linear case u(x, t) = 4t + i exp k 0 x + k0 2t ) k 0 = 8, Ω = [ 5, 5], 4t + i N = 1024, t = t=0 t=0.15 t=0.30 t=0.45 Evolution of u u x

46 2D linear case Explicit solution (2D) u(x 1, x 2, t) = i ( i 4t exp i x2 1 + x ix ) it. i 4t Finite Element Approximation (P 1 ): Ω i = D(0, 10), 3278 triangles, t = t=0.25 t=0.35 t=0.50

47

48 2D linear and nonlinear case with potentials Linear, Circular domain, V (x, y) = 5(x 2 + y 2 ) Linear, Mediator shaped domain, V (x, y) = 5 x 2 + y 2 Nonlinear, Circular domain, V (x, y) = 0 Nonlinear, Circular domain, V (x, y) = x 2 + y 2