High frequency wave propagation in non-uniform regular discrete media

Transcription

1 High frequency wave propagation in non-uniform regular discrete media Aurora MARICA BCAM - Basque Center for Applied Mathematics Derio, Basque Country, Spain Workshop of the ESF project OPTIMIZATION WITH PDE CONSTRAINTS Prague, Charles University December 09, 2011 joint work with Enrique Zuazua zuazua@bcamath.org Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

2 Problem formulation Transport and wave equation on R: ϱ(yu t(y, t u y (y, t = 0, y R, t > 0, u(y, 0 = u 0 (y. (1 ρ(yu tt(y, t (σ(yu y y (y, t = 0, y R, u(y, 0 = u 0 (y, u t(y, 0 = u 1 (y, y R. (2 Energy conserved in time: E ϱ(u 0 := 1 u(y, t 2 ϱ(y dy and E ρ,σ(u 0, u 1 := 1 (ρ(y u t(y, t 2 + σ(y u y (y, t 2 dy. 2 2 R R ϱ C 0,1 (R, (1 can be uniquely solved as u(y(t, t = u 0 (y, where ẏ(t = 1, t > 0, y(0 = y R. (3 ϱ(y(t ũ(ỹ, t = u(y, t, with ỹ = H(y and H (y = ϱ(y, (1 becomes ũ t(ỹ, t ũỹ (ỹ, t = 0, ỹ R, t > 0, ũ(ỹ, 0 = ũ 0 (ỹ := u 0 (H 1 (ỹ, (4 for which the solution is ũ(ỹ, t = ũ 0 (ỹ + t = u 0 (H 1 (ỹ + t and then the solution of (1 is given by u(y, t = u 0 (H 1 (H(y + t. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

3 Finite difference approximations Let h > 0 be the mesh size parameter, g : R R be an increasing function on R, G h := {x j := jh, j Z} and G h g := {g j := g(x j, j Z} the uniform grid of size h of R and the non-uniform one obtained by transforming the uniform one through the map g. Finite difference semi-discretization of (1 and (2 on the non-uniform grid G h g : and ϱ(g j u j,t (t u j+1(t u j 1 (t g j+1 g j 1 = 0, j Z, u j (0 = u 0 j, j Z (5 ρ(g j u j,tt (t σ(g j+1/2 u j+1(t u j (t g j+1 g j Energy is conserved in time σ(g j 1/2 u j (t u j 1 (t g j g j 1 g j+1 g j 1 2 = 0, u j (0 = u 0 j, u j,t(0 = u 1 j. (6 E h ϱ,g (u h,0 := h j R ϱ(g j h g j u j (t 2 and E h ρ,σ,g (u h,0, u h,1 := h [ h g j ρ(g j u j,t (t 2 + σ(g j+1/2 2 h,+ h,+ u j (t 2]. g j Z j Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

4 ( u h (t u(g(x h, t, u h t (t ut(g(xh, t l 2 hc(t, g, ρ, σ, u 0, u 1. (8 Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24 Convergence result Proposition ϱ C 1 (R s.t. 0 < ϱ ϱ(y ϱ + and ϱ (y ϱ +, u 0 Cc 2 (R in (1, g C 1 (R s.t. 0 < g g (x g + <, (5 with data u 0,h := (u 0 (g j j Z is convergent of order O(h for the transport equation u h (t u(g(x h, t l 2 Cth. (7 h Proposition ρ C 1 (R, σ C 2 (R s.t. 0 < ρ ρ(y ρ +, ρ (y ρ + d, 0 < σ σ(y σ +, σ (y σ + dd, (u 0, u 1 H k H k 1 (R, with k 3 + 1/2, in (2 g C 2 (R s.t. 0 < g d g (x g + d and g (x g + dd (6 with data (u h,0, u h,1 := (u 0 (g j, u 1 (g j j Z is convergent of order O(h for the wave equation

5 Aim: analyze the discrete versions of the observability inequality for (2: T E σ,ρ(u 0, u 1 C(T E σ,ρ,ω (u(, t, u t(, t dt, (9 0 for all solutions u of (2 with initial data (u 0, u 1 H := Ḣ 1 L 2 (R. Here Ω := R \ ( 1, 1 and E ρ,σ,ω (f 0, f 1 is the energy concentrated in Ω given below: E ρ,σ,ω (f 0, f 1 := 1 (σ(y fy 0 (y 2 + ρ(y f 1 (y 2 dy. 2 Ω Pathologies and remedies for numerical approximations of waves on uniform media Ervedoza, Zuazua, The wave equation: control and numerics, CIME Subseries, Springer. M., Zuazua, Localized solutions and filtering mechanisms for the DG semi-discretizations of the 1 d wave equation, CRAS, Rays of Geometric Optics: Bardos, Lebeau, Rauch, Sharp sufficient conditions for observation, control and stabilization of waves from the boundary, SIAM J. Cont. Optim., Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

6 Rays of GO - continuous and discrete on uniform meshes cases Continuous rays: x(t = x ± t Discrete rays on uniform meshes: x(t = x ± tω h (ξ 0, where ω h (ξ := 2 sin(ξh/2/h. (a continuous (b discrete, ξ 0 = π/2h (c discrete, ξ 0 = 2π/3h Nπ 0 0 N (d Continuous (blue and discrete (red dispersion relations Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

7 Known facts on non-uniform meshes I RESOLVENT ESTIMATES Ervedoza, Spectral conditions for admissibility and observability of wave systems, Numer. Mathematik, The observability inequality for general finite element semi-discretizations of the wave equation corresponding to a convergent approximation of order h θ of the Laplacian holds uniformly as h 0 in a class of truncated solutions generated by eigenvalues of order (ɛ/h θ 2, for some ɛ > 0, which does not include the critical scale 1/h 2 appearing in the numerical approximations on uniform meshes. MIXED FINITE ELEMENTS METHODS Ervedoza, On the mixed finite element method for the 1 d wave equation on non-uniform meshes, ESAIM:COCV, The numerical scheme: g j+1 g j (u j+1 4 (t+u j (t+ g j g j 1 4 (u j (t+u j 1 (t u j+1(t u j (t g j+1 g j u j (t u j 1 (t g j g j 1 = 0. Eigenvalues λ: kπ N 2 = arctan λ(g j+1 g j, 1 k N. 2 j=0 Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

8 Known facts on non-uniform meshes II SPECTRAL DISTRIBUTION OF LOCALLY TOEPLITZ SEQUENCE MATRICES Beckerman, Serra-Capizzano, On the asymptotic spectrum of FEM matrix sequences, SINUM, Tilli, Locally Toeplitz sequences: spectral properties and applications, Lin. Alg. Appl., (a(xu x x = b(x, x (0, 1 Szegö Theorem. Uniform mesh + constant coefficients. T N (ω = Toeplitz matrix whose diagonals are Fourier coefficients ω j (0 j N of ω : ( π, π C and (λ j 1 j N are the eigenvalues of T N (ω. Then F C b c (R: 1 lim N N N F (λ j = 1 π F (ω(ξ dξ. 2π j=1 π Uniform mesh + variable coefficients. T N (ω, a = locally Toeplitz matrix, a : (0, 1 R, then 1 N lim F (λ j = 1 π 1 F (ω(ξa(x dx dξ. N N 2π j=1 π 0 Non-uniform mesh + variable coefficients. Scheme generating a Toeplitz matrix whose diagonals are generated by ω and on a non-uniform mesh g locally Toeplitz matrix T N (ω, a, y s.t. 1 lim N N N F (λ j = 1 π 1 2π j=1 π 0 ( a(g(x F g (x 2 ω(ξ dx dξ. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

9 Wigner transform Continuous Wigner transform at the scale ɛ of f : W ɛ [f ](x, ξ := 1 f (x ɛz/2f (x + ɛz/2 exp(iξz dz. (10 2π R Singular phenomena: Concentration: f k (x = k 1/2 a(k(x x 0 Oscillation in the direction ξ 0 : g k (x = a(x exp(ikxξ 0. Defect measure: ν[f k ](x = a 2 2 δx 0 and ν[g k ](x = a(x 2 dx. Wigner measure: w ɛ [f k ](x, ξ a 2 2 δx 0 (x δ 0(ξ and w ɛ [g k ](x, ξ a(x 2 dx δ 0 (ξ for ɛk << 1 w ɛ [f k ](x, ξ δ x0 (x 1 2π â(ξ 2 dξ and w ɛ [g k ](x, ξ a(x 2 dx δ ξ0 (ξ as ɛk = 1 in S (R x R ξ. Discrete Wigner transform at scale ɛ: ( W ɛ [f h xm ] 2, ξ := 1 2π n m f m n f m+n exp(iξx n. 2 2 Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

10 Theorem ϱ in (1 s.t. 1/ϱ C 1,1 (R and u 0 = u 0,ɛ in (1 depending on the parameter ɛ > 0 s.t. E ϱ(u 0,ɛ is bounded as ɛ 0, a positive Radon measure µ = µ(x, t, ξ s.t. w ɛ [v ɛ ](y, t, ξ µ(y, t, ξ weakly star in S (R y R ξ uniformly in t. Moreover, µ satisfies the equation µ t(y, t, ξ = (1/ϱ(yµ y (y, t, ξ (1/ϱ (yξµ ξ (y, t, ξ, µ(y, 0, ξ = µ 0 (y, ξ. (11 Here, v ɛ (y, t := ϱ(yu ɛ (y, t. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

11 Proof of Theorem 1 Step I. Weak convergence of the Wigner transforms w ɛ [v ɛ ]: w ɛ (y, t, ξ µ(y, t, ξ weakly star in S (R y R ξ as ɛ 0, t 0, (12 where µ is a positive Radon measure. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

12 Proof of Theorem 1 Step I. Weak convergence of the Wigner transforms w ɛ [v ɛ ]: w ɛ (y, t, ξ µ(y, t, ξ weakly star in S (R y R ξ as ɛ 0, t 0, (12 where µ is a positive Radon measure. Step II. Equation satisfied by the Wigner transform w ɛ : { w ɛ t (y, t, ξ = K ɛ 1,c (y, w ɛ (y, t, ξ + K ɛ 1,c (y, w ɛ y (y, t, ξ + K ɛ 2,c (y, ( iξw ɛ (y, t, ξ, w ɛ (y, 0, ξ = w ɛ [v 0,ɛ ](y, ξ, where c(y := 1/ϱ(y. (13 Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

13 Proof of Theorem 1 Step I. Weak convergence of the Wigner transforms w ɛ [v ɛ ]: w ɛ (y, t, ξ µ(y, t, ξ weakly star in S (R y R ξ as ɛ 0, t 0, (12 where µ is a positive Radon measure. Step II. Equation satisfied by the Wigner transform w ɛ : { w ɛ t (y, t, ξ = K ɛ 1,c (y, w ɛ (y, t, ξ + K ɛ 1,c (y, w ɛ y (y, t, ξ + K ɛ 2,c (y, ( iξw ɛ (y, t, ξ, w ɛ (y, 0, ξ = w ɛ [v 0,ɛ ](y, ξ, where c(y := 1/ϱ(y. (13 Step III. Let us multiply equation (13 by a smooth function a S(R y R ξ, integrate in R 2, use Parseval identity in the ξ variable to obtain ɛ t (y, t, ξa(y, ξ dy dξ = (14 R 2 w = 1 [ K 2π 1,c ɛ (y, zŵ ɛ (y, t, z + K 1,c ɛ (y, z y ŵ ɛ (y, t, z + K 2,c ɛ (y, z z ŵ ɛ (y, t, z ] â(y, z dy dz. R 2 Here, denotes the Fourier transform in the ξ variable. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

14 Equation (14 can be written as: ɛ wt (y, t, ξa(y, ξ dy dξ = where (a c ɛ (y, ξ = 1 2π R 2 R R 2 w ɛ (y, t, ξ(a c ɛ (y, ξ dy dξ, (15 [ K ɛ 1,c (y, zâ(y, z y ( K ɛ 1,c (y, zâ(y, z z ( K ɛ 2,c (y, zâ(y, z] exp(iξz dz. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

15 Equation (14 can be written as: ɛ wt (y, t, ξa(y, ξ dy dξ = where (a c ɛ (y, ξ = 1 2π R 2 R R 2 w ɛ (y, t, ξ(a c ɛ (y, ξ dy dξ, (15 [ K ɛ 1,c (y, zâ(y, z y ( K ɛ 1,c (y, zâ(y, z z ( K ɛ 2,c (y, zâ(y, z] exp(iξz dz. (a c ɛ (y, ξ (a c 0 (y, ξ = y (c(ya(y, ξ + c (y ξ (ξa(y, ξ (16 strongly in S(R y R ξ as ɛ 0. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

16 Equation (14 can be written as: ɛ wt (y, t, ξa(y, ξ dy dξ = where (a c ɛ (y, ξ = 1 2π R 2 R R 2 w ɛ (y, t, ξ(a c ɛ (y, ξ dy dξ, (15 [ K ɛ 1,c (y, zâ(y, z y ( K ɛ 1,c (y, zâ(y, z z ( K ɛ 2,c (y, zâ(y, z] exp(iξz dz. (a c ɛ (y, ξ (a c 0 (y, ξ = y (c(ya(y, ξ + c (y ξ (ξa(y, ξ (16 strongly in S(R y R ξ as ɛ 0. Indeed, y α ( K α,c ɛ (y, zâ(y, z = y α K ɛ α,c (y, zâ(y, z + K α,c ɛ (y, z y α â(y, z, for all α {1, 2}, with y 1 = y and y 2 = z. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

17 Equation (14 can be written as: ɛ wt (y, t, ξa(y, ξ dy dξ = where (a c ɛ (y, ξ = 1 2π R 2 R R 2 w ɛ (y, t, ξ(a c ɛ (y, ξ dy dξ, (15 [ K ɛ 1,c (y, zâ(y, z y ( K ɛ 1,c (y, zâ(y, z z ( K ɛ 2,c (y, zâ(y, z] exp(iξz dz. (a c ɛ (y, ξ (a c 0 (y, ξ = y (c(ya(y, ξ + c (y ξ (ξa(y, ξ (16 strongly in S(R y R ξ as ɛ 0. Indeed, y α ( K α,c ɛ (y, zâ(y, z = y α K ɛ α,c (y, zâ(y, z + K α,c ɛ (y, z y α â(y, z, for all α {1, 2}, with y 1 = y and y 2 = z. Moreover y K ɛ 1,c = K ɛ 1,c, z K ɛ 2,c = K ɛ 1,c, Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

18 Equation (14 can be written as: ɛ wt (y, t, ξa(y, ξ dy dξ = where (a c ɛ (y, ξ = 1 2π R 2 R R 2 w ɛ (y, t, ξ(a c ɛ (y, ξ dy dξ, (15 [ K ɛ 1,c (y, zâ(y, z y ( K ɛ 1,c (y, zâ(y, z z ( K ɛ 2,c (y, zâ(y, z] exp(iξz dz. (a c ɛ (y, ξ (a c 0 (y, ξ = y (c(ya(y, ξ + c (y ξ (ξa(y, ξ (16 strongly in S(R y R ξ as ɛ 0. Indeed, y α ( K α,c ɛ (y, zâ(y, z = y α K ɛ α,c (y, zâ(y, z + K α,c ɛ (y, z y α â(y, z, for all α {1, 2}, with y 1 = y and y 2 = z. Moreover y K ɛ 1,c = K 1,c ɛ, z K ɛ 2,c = K 1,c ɛ, 1 K 1, c ɛ (y, zâ(y, z exp(iξz dz c(ya(y, ξ 2π R Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

19 Equation (14 can be written as: ɛ wt (y, t, ξa(y, ξ dy dξ = where (a c ɛ (y, ξ = 1 2π R 2 R R 2 w ɛ (y, t, ξ(a c ɛ (y, ξ dy dξ, (15 [ K ɛ 1,c (y, zâ(y, z y ( K ɛ 1,c (y, zâ(y, z z ( K ɛ 2,c (y, zâ(y, z] exp(iξz dz. (a c ɛ (y, ξ (a c 0 (y, ξ = y (c(ya(y, ξ + c (y ξ (ξa(y, ξ (16 strongly in S(R y R ξ as ɛ 0. Indeed, y α ( K α,c ɛ (y, zâ(y, z = y α K ɛ α,c (y, zâ(y, z + K α,c ɛ (y, z y α â(y, z, for all α {1, 2}, with y 1 = y and y 2 = z. Moreover y K ɛ 1,c = K 1,c ɛ, z K ɛ 2,c = K 1,c ɛ, 1 K 1, c ɛ (y, zâ(y, z exp(iξz dz c(ya(y, ξ 2π R and 1 K 2, c ɛ 2π (y, zâz (y, z exp(iξz dz c (y ξ (ξa(y, ξ, R for all enough regular function c. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

20 Gérard, Markowich, Mauser, Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., Lions P.-L., Paul, Sur les mesures de Wigner, Rev. Matemática Iberoamericana, Schrödinger equation with potential ihψ h t = h2 ψ h xx + V (xψh, (x, t R R, is transformed into Liouville equation w t + ξw x V x w ξ = 0, which needs V C 1,1 (R for uniqueness. (y(t, ξ(t is the solution of the following Hamiltonian system: y (t = (1/ϱ(y(t, ξ (t = (1/ϱ (y(tξ(t, y(0 = y, ξ(0 = ξ. (17 p(y, t, ξ, τ := ϱ(yτ + ξ - the Hamiltonian associated to (1 whose null bi-characteristic lines are the solutions of { Ẏ (s = ξ p(y (s, t(s, Ξ(s, τ(s = 1, ṫ(s = τ p(y (s, t(s, Ξ(s, τ(s = ϱ(y (s, Ξ(s = y p(y (s, t(s, Ξ(s, τ(s = τϱ (Y (s, τ(s = tp = 0 (y(t, ξ(t = (Y (s, Ξ(s. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

21 High frequency wave propagation for the transport equation ϱ(g(x g(x + h g(x h ṽt h (x, t ṽ h (x + h, t ṽ h (x h, t = 0, ṽ h (x, 0 = ṽ 0,h (x. 2h 2h (18 ϱ h g (x := ϱ(g(x(g(x + h g(x h/2h ϱg (x := ϱ(g(xg (x. v h (x, t = ϱ h g (xṽ h (x, t satisfies the equation: v h t (x, t = Theorem v h (x+h,t ϱ h v (x h,t h g (x+h ϱ h g ϱ (x h g (x 2h ϱ h g (x h, v h (x, 0 = v 0,h (x := ϱ h g (xṽ 0,h (x. (19 For all ϱ in (1 and all g s.t. 1/ϱ g C 1,1 (R and all bounded v 0,h, there exists a positive Radon measure µ = µ(x, t, ξ defined on R R + R such that w h [v h ](x, t, ξ µ(x, t, ξ weakly star in S (R x D ([ π, π] uniformly in t. The measure µ satisfies the equation µ t = (1/ϱ g (xcos(ξµ x (1/ϱ g (xsin(ξµ ξ, µ(x, 0, ξ = µ 0 (x, ξ := lim h 0 w h [v 0,h ](x, ξ. (20 Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

22 (x(t, ξ(t is the solution of the following Hamiltonian system: x (t = (1/ϱ g (x(tcos(ξ(t, ξ (t = (1/ϱ g (x(tsin(ξ(t, x(0 = x, ξ(0 = ξ. (21 p(x, t, ξ, τ := ϱ g (xτ + sin(ξ - the Hamiltonian associated to the solution of (5 whose null bi-characteristic lines are the solutions of the following system Ẋ (s = ξ p(x (s, t(s, Ξ(s, τ(s = cos(ξ(s, ṫ(s = τ p(x (s, t(s, Ξ(s, τ(s = ϱ g (X (s, Ξ(s = x p(x (s, t(s, Ξ(s, τ(s = τϱ g (X (s, τ(s = tp(x (s, t(s, Ξ(s, τ(s = 0. (22 Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

23 Wigner measures for the continuous wave equation (w(y, t, w(y, t := ( ρ(yu t(y, t, σ(yu y (y, t satisfies: ( ( ( wt(y, t w(y, t 0 c(y = A, A := y + d(y w t(y, t w(y, t c(y y + e(y 0 σ(y c(y := ρ(y, d(y := σ (y σ(yρ 2 (y and e(y := σ(yρ(y Theorem 2ρ(y ρ(y,, (23 For each coefficient c C 1,1 (R and each initial data (w ɛ,0, w ɛ,0 in (23 bounded in (L 2 (R 2 as ɛ 0, there exists a positive Radon measure W(y, t, ξ defined on R R + R such that W ɛ [w ɛ ](y, t, ξ + W ɛ [ w ɛ ](y, t, ξ W(y, t, ξ weakly star in S (R y R ξ. Moreover, W can be split into two positive Radon measures as W = W + + W, where W ± := lim W ɛ,± is the Wigner ɛ 0 measures of w ɛ,± solving (25 with initial data w ɛ,0,± := (w ɛ,0 ± w ɛ,0 / 2. Each Wigner measure W ± satisfies the transport equation W ± t (y, t, ξ = c(yw± y (y, t, ξ ± c (yξw ± ξ (y, t, ξ, W± (y, 0, ξ = lim ɛ 0 W ɛ,± (y, 0, ξ. (24 w ± (y, t = (w(y, t ± w(y, t/ 2 satisfies: ( w + ( t (y, t w wt = (y, t à + ( (y, t c(y y + 1 w, à := 2 c (y 1 2 c(y (y, t 1 2 c(y c(y y 1 2 c (y (25 Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

24 Given a d d - matrix valued function Θ(y, ξ, define the pseudo-differential operator Θ(y, ɛ y : Θ(y, ɛ y f(y := 1 Θ(y, ɛξ f(ξ exp(iξy dξ. (26 2π R Set w(y, t := (w(y, t, w(y, t and Θ(y, ξ := Θ 1 (y, ξ + ɛθ 0 (y, ξ, where ( ( 0 c(yiξ 0 d(y Θ 1 (y, ξ := and Θ c(yiξ 0 0 (y, ξ := e(y 0 Using this notation, system (23 can be written as w t(y, t = 1 ( 1 ɛ Θ(y, ɛ y w(y, t = ɛ Θ 1(y, ɛ y + Θ 0 (y, ɛ y w(y, t. (27 The matrix Θ 1 (y, ξ admits the Fourier decomposition Θ 1 (y, ξ = i (y, ξλ(y, ξ( (y, ξ, where ( λ + c,ξ (y, ξ 0 Λ(y, ξ := 0 λ, (y, ξ := 1 ( 1 1, (28 c,ξ (y, ξ and λ ± c,ξ (y, ξ := ±c(yξ. Define the projectors ± = ± (y, ξ ( := 0 0 = 1 ( ( 1 1 and = 0 1 In this way, the matrix Θ 1 (y, ξ can be decomposed as Moreover, the following identities hold:. = 1 ( Θ 1 (y, ξ = iλ + c,ξ (y, ξ + + iλ c,ξ (y, ξ. (29 ± ± = ±, ± = 0, + + = I 2. (30 Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24.

25 Expression of the Fourier transform in ξ of W ɛ [w ɛ ](y, t, ξ, Ŵɛ [w ɛ ](y, t, z: Ŵ ɛ t [w ɛ ](y, t, z = 1 ɛ Θ(y, ɛ y wɛ( y ɛz ( 2, t w ɛ( y + ɛz 2, t (31 + w ɛ( y ɛz 2, t 1 ( Θ(y, ɛ y w ɛ( y + ɛz. ɛ 2, t Multiply (31 by â(y, z/2π: where A ɛ = 1 2 (2π 2 ɛ R 4 and B ɛ = 1 2 (2π 2 ɛ R 4 1 2π Ŵɛ t [wɛ ], â S (R 2,S(R 2 = Aɛ + B ɛ, (32 1 ɛ Θ(2y x ɛz, ζwɛ (x, t (w ɛ (2y x, t exp ( 2iζ ɛ ( y x ɛz â(2y x ɛz 2 2, z w ɛ (x, t (w ɛ (2y x, t ( 1 ( 2iζ ( ɛ Θ(x+ɛz, ζ exp y x ɛz â(x+ ɛz ɛ 2 2, z dx dy 2y x ɛz = y + y 1, x + ɛz = y y 1, 2y x ɛz/2 = y + y 2 and x + ɛz/2 = y y 2, with y 1 = y x ɛz and y 2 = y x ɛz/2. Taylor expansions of Θ(y ± y 1, ζ and â(y ± y 2, z Θ(y ± y 1, ζ = Θ(y, ζ ± y 1 Θ y (y, ζ + y 2 1 R± Θ and â(y ± y 2, z = â(y, z ± y 2 â y (y, z + y 2 2 R± a. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

26 W ɛ t [w ɛ ] = Θ 1W ɛ [w ɛ ] W ɛ [w ɛ ]Θ 1 + Θ 0 W ɛ [w ɛ ] + W ɛ [w ɛ ]Θ 0 ɛ + 1 2i ({Θ 1, W ɛ [w ɛ ]} {W ɛ [w ɛ ], Θ 1 } (33 1 2i (Θ 1,yζW ɛ [w ɛ ] + W ɛ [w ɛ ]Θ 1,yζ + ɛr ɛ, where R ɛ is bounded in (S (R 2 4. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

27 The principal symbol associated to numerical scheme for the wave equation: ( (x, t, ξ, τ := g (xρ(g(xτ sin 2 ξ σ(g(x 2 g (x. (34 The null bi-characteristic lines associated to this principal symbol are the solutions of Ẋ (s = ξ = 2 sin(ξ(s σ(g(x (s g (X (s Ξ(s = x = (g ( ρ(g( (X (s 4 sin 2 ( Ξ(s 2, ṫ(s = τ = g (X (sρ(g(x (sτ, (X (s, τ(s = t = 0, ( σ(g( g ( subjected to the initial data (X (0, t(0, Ξ(0, τ(0 = (x 0, 0, ξ 0, τ 0 s.t. (x 0, 0, ξ 0, τ 0 = 0. (x ± (t, ξ ± (t := (X (s, Ξ(s is solution of (35 for τ ± 0 = ±2 sin(ξ(s/2c g (X (s, with c g (x := σ(g(x/ρ(g(x/g (x. (x ± (t, ξ ± (t is the solution of the following Hamiltonian system: ( ξ (x ± (t = c g (x ± ± (t (t cos, (ξ ± (t = ±c g 2 (x± (t2 sin ( ξ ± (t α h 1 (x := h g(xρ(g(x 1 ρ(g(xg (x, σ(g(x + h/2 βh (x := h,+ g(x γ h (x := α h (xβ h (x c g (x := 1 σ(g(x g (x ρ(g(x, δ h,+ (x c g (x and δ h, (x σ(g(x g (x 2 (35, x(0 = x 0, ξ(0 = ξ 0. σ(g(x g (x, δh,± (x := β h (x h,+ α h (x±α h (x h, β h (x. ( 1 (x. ρ(gσ(g Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

28 Set w j (t := v j (t/α j and w j (t := β j h,+ v j (t. (w h (t, w h (t are the solution of w j (t = γ j h, w j (t + α j ( h, β j w j 1 (t, w j (t = γ j h,+ w j (t + β j ( h,+ α j w j+1, (36 (36 can be also written in terms of the two pseudo-differential operators Θ h 1 (x, h x and Θ h 0 (x, h x generated by the matrices ( 0 γ Θ 1 (x, ξ := h (x(1 exp( iξ γ h (x(exp(iξ 1 0 and as follows ( Θ 0 (x, ξ := ( w h t (t w h t (t 0 α h (x h, β h (x exp( iξ β h (x h,+ α h (x exp(iξ 0 = 1 h Θh 1 (x, h x ( w h (t w h (t + Θ h 0 (x, h x ( w h (t w h (t. (37 The matrix Θ 1 (x, ξ admits the spectral decomposition Θ 1 (x, ξ = i (x, ξλ(x, ξ (x, ξ, where ( λ + (x, ξ 0 γ Λ(x, ξ := h,ω 0 λ, (x, ξ := 1 ( 1 1, (x, ξ γ h,ω 2 exp(iξ/2 exp(iξ/2 with λ ± γ h,ω (x, ξ = ±γh (x2 sin(ξ/2. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

29 Set ( w h,+ (t w h, (t ( =,h w (x, h x h (t w h (t. (38 Theorem For coefficient ρ, σ and non-uniform grids obtained by transformations g s.t. c g C 1,1 (R and for each initial data (w h,0, w h,0 in (36 bounded in l 2 as h 0, there exists a positive Radon measure W = W(x, t, ξ defined on R R + [ π, π] such that W h [w h ](x, t, ξ + W h [ w h ](x, t, ξ W(x, t, ξ weakly star in S (R x D ([ π, π]. Moreover, the measure W can be decomposed as W = W + + W, where W ± = lim h 0 W h [w h,± ] satisfies: W ± t (x, t, ξ = cg (xcos(ξ/2w± x (x, t, ξ ± c g (x2 sin(ξ/2w± ξ (x, t, ξ. (39 Macìa, Propagación y control de vibraciones en medios discretos y continuos, PhD. Thesis, Macìa, Wigner measures in the discrete setting: high frequency analysis of sampling and reconstruction operators, SIAM J. Math. Anal., Macìa, Zuazua, On the lack of observability for wave equations: a Gaussian beam approach, Asympt. Anal., Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

30 Numerical simulations Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

31 Conclusions and open problems In this talk: We give a meaning to the notion of rays of Geometric Optics by constructing appropriate transport equations. Open problems: To solve the Hamiltonian systems, we need ρ(g(xg (x C 1,1 (R. Study the case when the non-homogeneity of the grid is less regular. Adapt the multiplier techniques to prove the observability inequality for the non-uniform mesh case (a posteriori error estimates techniques. Adapt the filtering techniques (the bi-grid ones to remedy the pathological effects of the high-frequency spurious solutions. Study more sophisticated methods for the wave equation (DG ones, higher order ones on non-uniform meshes. The multi-d case. Dispersive estimates for the Schrödinger equation on non-uniform meshes. Meshes which are given randomly. Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24

32 Conclusions and open problems In this talk: We give a meaning to the notion of rays of Geometric Optics by constructing appropriate transport equations. Open problems: To solve the Hamiltonian systems, we need ρ(g(xg (x C 1,1 (R. Study the case when the non-homogeneity of the grid is less regular. Adapt the multiplier techniques to prove the observability inequality for the non-uniform mesh case (a posteriori error estimates techniques. Adapt the filtering techniques (the bi-grid ones to remedy the pathological effects of the high-frequency spurious solutions. Study more sophisticated methods for the wave equation (DG ones, higher order ones on non-uniform meshes. The multi-d case. Dispersive estimates for the Schrödinger equation on non-uniform meshes. Meshes which are given randomly. Thank you very much for your attention! Aurora Marica (BCAM Wave propagation heterogeneous media OPTPDE, Prague - 09/12/ / 24