A stroboscopic averaging technique for highly-oscillatory Schrödinger equation

Transcription

1 A stroboscopic averaging technique for highly-oscillatory Schrödinger equation F. Castella IRMAR and INRIA-Rennes Joint work with P. Chartier, F. Méhats and A. Murua Saint-Malo Workshop, January 26-28, 2011

2 Goal of the present study Numerical schemes and asymptotic study for nonlinear Schrödinger equations of the form i t ψ ε (t, x) = H ( x ε ψε (t, x) + F ψ ε (t, x) 2) ψ ε (t, x), where the Hamiltonian H x is the harmonic oscillator (x R d ) H x = 1 2 x x 2. and F is a given nonlinear function. Context: Bose-Einstein condensates H x /ε represents strong confinement in x. nonlinear term describes bosons interactions.

3 Goal of the present study (2) References (in that context): Bao, Markowich, Schmeiser, Weishäupl (M3AS, 2005), Ben Abdallah, Méhats (JMPA, 2005), Ben Abdallah, Méhats, Schmeiser, Weishäupl (SIAM, 2005). See also, in the context of fluid mechanics: Grenier (JMPA, 1997), Schochet (JDE, 1994), Métivier, Schochet (JDE 2003), and, for other Schrödinger-like or Vlasov-like equations: Bidégaray, Castella, Degond (DCDS 2004), Castella, Degond, Goudon (JSP 2006).

4 Rough analysis (and wrong approach) Linear part: Eigenvalues of H x : {E n = n n d + d 2 ; n = (n 1,..., n d ) N d }, Eigenfunctions: explicitely known χ n (x) s (n N d ), Hermite polynomial Gaussian. Project ψ ε (t, x) = n ψ ε n (t)χ n(x), and write, say if F( ψ 2 )ψ = ψ 2 ψ, i t ψn ε (t) = E n ε ψε n (t) + A n,p,q,r ψp ε (t) ψq ε (t)ψε r (t) p,q,r ( ) with A n,p,q,r = ψp ε (t) ψq ε (t)ψε r (t). R d

5 Rough analysis (and wrong approach) (2) Filter out the oscillatory term and introduce ( φ ε n (t, x) = exp +it E ) n ψ ε (t, x). ε It satisfies i t φ ε n(t) = p,q,r A n,p,q,r exp ( it E ) n + E p E q E r φ ε ε p(t) φ ε q(t)φ ε r (t). System of the form ( ) t t u ε = G ε, uε, with G periodic (E n s are half-integers) and u ε = (φ ε 0,φε 1,...) (infinite, nonlinearly coupled system).

6 Rough analysis (and wrong approach) (3) Questions How to numerically average out the ODE ( ) t t u ε = G ε, uε? Answer inspired by Chartier, Murua, Sanz-Serna (ODE s). How to control the norms of u ε, and that of the nonlinear term G (i.e., typically φ ε and the sums p,q,r : it is not clear that the Sobolev smoothness n nα φ n 2 < + implies n nα p,q,r 2 < +, because of the An,p,q,r s)? Answer inspired by Ben Abdallah, Castella, Méhats (functional analytic framework).

7 Right approach and analytic framework Avoid projecting on the χ n s and write directly (Ben Abdallah, Castella, Méhats) ( φ ε (t, x) = exp +it H ) x ψ ε (t, x). ε ( ) t It satisfies i t φ ε (t, x) = G ε,φε (t, x), ( F(z) = F( z 2 ) z) where, for u = u(x) given, we have ( ) ( t G ε, u = exp +it H ) ( x F exp ε ( it H x ε ) ) u(x)

8 Right approach and analytic framework (2) Bounds on the nonlinear terms G(t/ε,φ ε ) or F(ψ ε )? When F 0, we have (take a large N N) i t ψ ε = H x ε ψε while = t H N x ψ ε 2 = 2Re L 2 H N x ( i H ) x ε ψε, Hx L N ψε = 0, 2 (H x commutes with Hx N and H x is self-adjoint) = Hx N ψ ε 2 const. L 2 t N x ψ ε 2 L 2 = O ( ) 1 ε = no uniform in ε bound on N x ψε 2 L 2.

9 Right approach and analytic framework (3) This is due to t N x ψ ε 2 = 2Re = O ( 1 ε N x = 2Re L 2 ). N x ( i x 2 ε ψε ( i H x ε ψε ), x N ψ ε ), x N ψ ε Only a uniform bound on H N x ψε L 2 (and not on N x ψε L 2) can be obtained in the general case. The good norm is u B N = u L 2 + H N x u L 2, instead of the Sobolev scale u H N = u L 2 + N x u L 2. L 2 L 2

10 Right approach and analytic framework (4) Note that : ( Hx N L φε = H N 2 x exp +it H ) x L ψ ε ε 2 ( = exp +it H ) x L ε HN x ψ ε (commutation) 2 = Hx N ψ ε L (self-adjointness) 2 Remaining question: we know ( u H N := u L 2 + x Nu L 2) F(ψ ε ) const. ( ψε H N), H N yet is it true that ( u B N := u L 2 + Hx Nu L 2) F(ψ ε ) const. ( ψε B N B N)?

11 Right approach and analytic framework (5) Answer (Ben Abdallah, Castella, Méhats, see also Helffer, Nier, and Bony, Chemin requires Weyl-Hörmander calculus for general Hamiltonians): Yes because of the following Theorem (H x = x + x 2 ) For any N, we have the equivalence of norms u L 2 + 2N x u L 2 + x 2N u L 2 u L 2 + ( 2 x + x 2 ) N u L 2 or, in other words u L 2 + 2N x u L 2 + x 2N u L 2 u B N.

12 Right approach and analytic framework (6) By standard nonlinear analysis (Gronwall + the above nonlinear estimates on F(u) B N + standard fixed point in the original PDE), this immediately implies the existence of T 0 > 0 such that ψ ε (t, x) B N, φ ε (t, x) B N const. (0 t T 0 ), and all nonlinear terms are well defined and uniformly bounded in the space B N there remains to average out, in the space B N, the equation ( ) t i t φ ε (t, x) = G ε,φε (t, x)

13 Highly oscillatory ODE s Problems of the form ẏ = dy dt = f(y, t/ε), y(0) = y 0, where : ε is a small parameter (the inverse of the frequency). f is a smooth vector-field, 2π-periodic in t/ε. Assumptions: a clear and explicit separation of the time-scales a periodic (and not only quasi-periodic at this stage) dependence in the fast time

14 Highly oscillatory ODE s (2) Appears in various forms: Standard form in KAM theory Two-scale expansions in quantum mechanics (Wentzel-Kramers-Brillouin, 1926) Magnus expansions (A. Iserles, 2002) Modulated Fourier expansions (D. Cohen, E. Hairer and Chr. Lubich, 2003) Theory has been gradually improved: Krylov and Bogoliubov (1934) : basic idea Bogoliubov and Mitropolski (1958) : rigorous statement for second order approximation and general scheme Perko (1969) : almost complete theory with error estimates for the periodic and quasi-periodic cases (see also the book Sanders, Verhulst and Murdock, 2007)

15 Highly oscillatory ODE s (3) Theorem (Perko, SIAM 1969 : Averaging of high-order) Under a smoothness assumption on f and a non-resonance condition on ω, given the system { y = εf(y,θ) R n, y(0) = y 0, θ = ω T d, θ(0) = θ 0, there exists a transformation from R n T d to R n such that y = U(Y,θ) = Y + εu 1 (Y,θ) ε k u k (Y,θ) Y = εf 1 (Y) ε k F k (Y), Y(0) = ξ. and y(τ) U(Y(τ),θ 0 + τω) Cε k for τ C/ε.

16 Highly oscillatory ODE s (4) The functions u i (and thus F i ) are not unique except F 1 F 1 (Y) = f(y,θ), j 1 [ 1 F k f ( ) j (Y,θ) = k! y k u i1,..., u ik u ] k Y F j k, k=1 i i k =j 1 F j (Y) = 1 (2π) d Fj (Y,θ)dθ, T d ω u j θ (Y,θ) = F j (Y,θ) F j (Y). For d = 1, one can impose u j (Y, 0) = 0: this is stroboscopic averaging, in the sense that U(Y, 2kπ) = Y.

17 An example For f(y,τ) = 1 + cos(τ)y, we have Φ τ τ 0 (y 0 ) = εe ε sin(τ) τ so that τ 0 e ε sin(s) ds + e ε(sin(τ) sin(τ 0)) y 0, Φ τ 0+2π τ 0 (y 0 ) = 2πενe ε sin(τ 0) + y 0, ν = 1 2π ( We see that Φ τ 0+2π τ 0 2π 0 e ε sin(τ) dτ. ) k (y0 ) coincides with the solution at points τ = 2kπ of the differential equation { Y (τ) = ενe ε sin(τ 0) Y(0) = y 0, which happens to be the modified equation we are looking for.

18 An example (2) Example 2.4 Exact solutions versus their averaged versions y and Y Time We search for a differential equation whose flow coincides at times that are multiple of 2π with the flow of the highly-oscillatory equation.

19 B-series expansions Mode-coloured B-series with time-dependent coefficients Consider the rescaled system expanded in Fourier coefficients { y (τ) = ε k Z eikτ f k (y), y(0) = y 0 θ (τ) = 1, θ(0) = θ 0 where the f k s are the Fourier coefficients of f. Chartier, Murua and Sanz-Serna [FOCM 2010] consider B-series of the form where y 0 + u T ε u α u(τ) σ u F u (y 0 ) T is a set of mode-decorated trees. F u (y 0 ) are associated derivatives of the f k s. σ u is a normalisation coefficient. α u (τ) are coefficients defining the series.

20 B-series expansions (2) Tree Order σ u F u k 1 1 f k l k = [ k ] l 2 1 f l f k k l = [ k, l ] m δ k,l f m(f k, f l ) m k k l [ l ] m = m 3 1 f mf l f k Table: Mode-decorated trees and their associated coefficients

21 B-series expansions (3) Nota : The general procedure is described as in Chartier, Murua, Sanz-Serna, FOCM 2010 Construction of averaged vector field F from εf(y, τ): 1 Compute the B-series expansion of the exact solution y(τ) = y 0 + u T ε u α u(τ) σ u F u (y 0 ) 2 Notice that α u (τ) = P u (τ,θ) θ=τ with P u (τ,θ) = a u kl τ l e ikθ 3 Define averaged (or frozen) solution by ᾱ u (τ) = P u (τ, 0) Y(τ) = y 0 + u T ε u ᾱu(τ) σ u F u (y 0 ) 4 Define averaged vector field εf(y,τ) such that { Y (τ) = εf(y(τ),τ) Y(0) = y 0 5 Notice that F is autonomous and Hamiltonian as soon as f is Hamiltonian.

22 The averaged vector field is autonomous The vector field corresponding to Y(τ) = Φ τ τ 0 (y 0 ) writes F(Y,τ) = ( Φ τ τ 0 τ ( Φτ τ0 ) 1)(Y) = u T where β(τ) = ᾱ 1 (τ) ᾱ (τ). Since Y(τ j+k ) = ε β u u (τ) F u (Y) σ u ( Φ τ 0+2π τ 0 ) j+k (y0 ): ᾱ u (τ j + τ k ) = ᾱ u (τ k ) ᾱ u (τ j ) = ᾱ u (τ j ) ᾱ u (τ k ), j, k = 0, ±1, ±2,... and by an interpolation argument k, τ, ᾱ u (τ + τ k ) = ᾱ u (τ k ) ᾱ u (τ). Differentiating and taking the value at τ = 0 gives k, ᾱ (τ k ) = ᾱ(τ k ) ᾱ (0), so that by interpolation once again, τ, ᾱ (τ) = ᾱ(τ) ᾱ (0) i.e. τ, ᾱ 1 (τ) ᾱ (τ) = ᾱ (0).

23 The averaged vector field is Hamiltonian If α is symplectic, the B-series ᾱ remains symplectic Since α u (τ) = P u (τ,τ) with P u (τ,θ) := a u lk τ l e ikθ we see that the symplecticity relation τ, α u v (τ) + α v u (τ) = α u (τ)α v (τ) implies τ, θ, P u v (τ,θ) + P v u (τ,θ) = P u (τ,θ)p v (τ,θ) and hence τ, ᾱ u v (τ) + ᾱ v u (τ) = ᾱ u (τ)ᾱ v (τ)

24 The numerical scheme The idea is to solve the averaged equation at two levels, in the spirit of Heterogeneous Multiscale Methods: Approximate the averaged vector field F by central differences of the form F(Y) 1 4π (Φ2π 0 (Y) Φ 2π 0 (Y)) with a numerical method with constant stepsizes (micro-steps). Solve the averaged equation by a numerical method with possibly variable stepsizes (macro-steps) Nota : in the formulation ẏ = f(t/ε, y), this gives rise to micro-steps δt ε alternating with macro-steps ε t 1.

25 p Stroboscopic averaging for nonlinear Schrödinger equations The numerical scheme (2) Example (Van Der Pol oscillator) 2.5 Van Der Pol oscillator { q = p ṗ = q + ε(1 q 2 )p q which, after the change of variables writes q = cos(t)x + sin(t)y p = sin(t)x + cos(t)y j ẋ = sin (t) ε `1 (cos (t) x + sin (t) y) 2 ( sin (t) x + cos (t) y) ẏ = cos (t) ε `1 2 (cos (t) x + sin (t) y) ( sin (t) x + cos (t) y)

26 Stroboscopic averaging for nonlinear Schro dinger equations The numerical scheme (3) Van Der Pol oscillator p q Figure: Exact solution (red) and numerical solution obtained by SAM (blue)

27 The numerical scheme : references References P. Chartier, J.M. Sanz-Serna and A. Murua, Higher-order averaging, formal series and numerical integration I: B-series, FOCM, M.P. Calvo, P. Chartier, J.M. Sanz-Serna and A. Murua, A stroboscopic numerical method for highly oscillatory problems, submitted. P. Chartier, J.M. Sanz-Serna and A. Murua, Higher-order averaging, formal series and numerical integration II: the multi-frequency case, in preparation.

28 Application to the considered nonlinear Schrödinger equations Theorem The Perko formulae apply in our context, when conveniently rephrased in the B N spaces: one can define functions u j, Y j as in Perko, and averaging can be performed at any order, to provided φ ε K (t, x) s such that φ ε K (t, x) = φε (t, x) + O(ε K ) in B N, whenever 0 t T 0. Numerical counterpart The above described numerical scheme, when conveniently adapted, provides the following results.

29 Application to the considered nonlinear Schrödinger equations (2) In the case of i t ψ ε = 1 ε x + ψ ε 2 ψ ε x [0, 1] with periodic boundary conditions, t [0, 2/2π] error eps=4.4e 2 eps=1.1e 2 eps=2.8e eps=6.9e 4 eps=1.7e 4 eps=8.6e time step Figure: direct Strang splitting (order 2 if ε 1)

30 Application to the considered nonlinear Schrödinger equations (3) error eps=4.4e 2 eps=1.1e 2 eps=2.8e 3 eps=6.9e 4 eps=1.7e 4 eps=8.6e time step Figure: direct order 4 splitting

31 Application to the considered nonlinear Schrödinger equations (4) error eps=4.4e 2 eps=1.1e 2 eps=2.8e 3 eps=6.9e 4 eps=1.7e 4 eps=8.6e time step Figure: direct order 6 splitting

32 Application to the considered nonlinear Schrödinger equations (5) eps=2.2e 2 eps=5.5e 3 eps=1.4e 3 eps=3.5e 4 eps=8.6e error micro step Figure: Stroboscopic averaging constant numerical cost, various values of ε

33 Application to the considered nonlinear Schrödinger equations (6) energy time Figure: Energy conservation.

34 Application to the considered nonlinear Schrödinger equations (7) u p p=1 p=2 p=3 p=4 p=5 p=6 p=7 p= time Figure: Evolution of modes 1 to 8 in the cubic NLS: modes 1 and 5 feed mode 7 (and only this mode).