www.bcamath.org Wave propagation in discrete heterogeneous media Aurora MARICA marica@bcamath.org BCAM - Basque Center for Applied Mathematics Derio, Basque Country, Spain Summer school & workshop: PDEs, optimal design and numerics - IV edition Benasque, Centro de Ciencias Pedro Pascual August 3, 20 Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 /
Problem formulation Finite difference approximation of the homogeneous wave equation on a non-uniform grid y = (y j 0 j N+ of (0, : u j+ (t u j (t u j u j (t u j (t y (t j+ y j y j y j y j+ y j = 0, ( 2 u 0 (t = u N+ (t = 0, u j (0 = uj 0, u j (0 = u j, j N. Aim: analyze the observability inequality for (: where h n uh (t = u N (t/( y N and E h (u h,0, u h, = 2 j= T E h (u h,0, u h, C h (T n h uh (t 2 dt, 0 N y j+ y j u j 2 (t 2 + 2 N (y j+ y j u j+ (t u j (t 2 y j=0 j+ y. j Pathologies and remedies for numerical approximations of waves on uniform media y = x = [0 : h : ], h = /(N + : Ervedoza, Zuazua, The wave equation: control and numerics, CIME Subseries, Springer. Rays of Geometric Optics: Bardos, Lebeau, Rauch, Sharp sufficient conditions for observation, control and stabilization of waves from the boundary, SIAM J. Cont. Optim., 992. Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 2 /
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 Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 3 /
Known facts on non-uniform meshes I RESOLVENT ESTIMATES Ervedoza, Spectral conditions for admissibility and observability of wave systems, Numer. Mathematik, 2009. 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 /h 2 appearing in the numerical approximations on uniform meshes. MIXED FINITE ELEMENTS METHODS Ervedoza, On the mixed finite element method for the d wave equation on non-uniform meshes, ESAIM:COCV, 200. The numerical scheme: y j+ y j (u j+ 4 (t+u j (t+ y j y j 4 (u j (t+u j (t u j+(t u j (t y j+ y j u j (t u j (t y j y j = 0. Eigenvalues λ: kπ N 2 = arctan λ(y j+ y j, k N. 2 j=0 Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 4 /
Known facts on non-uniform meshes II SPECTRAL DISTRIBUTION OF LOCALLY TOEPLITZ SEQUENCE MATRICES Beckerman, Serra-Capizzano, On the asymptotic spectrum of finite element matrix sequences, SINUM, 2007. Tilli, Locally Toeplitz sequences: spectral properties and applications, Lin. Alg. Appl., 998. 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 j N are the eigenvalues of T N (ω. Then F C b c (R: lim N N N F (λ j = π F (ω(ξ dξ. 2π j= π Uniform mesh + variable coefficients. T N (ω, a = locally Toeplitz matrix, a : (0, R, then N lim F (λ j = π F (ω(ξa(x dx dξ. N N 2π j= π 0 Non-uniform mesh + variable coefficients. For (a(xu x x = b(x, x (0, discretized using a scheme generating a Toeplitz matrix whose diagonals are generated by ω and on a non-uniform mesh y : (0, (0,, we obtain the locally Toeplitz matrix T N (ω, a, y s.t. lim N N N F (λ j = π 2π j= π 0 ( a(y(x F y (x 2 ω(ξ dx dξ. Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 5 /
Our results based on pseudo-differential calculus The d transport equation: ρ(xu ɛ t (x, t u ɛ x (x, t = 0, x R, t > 0, u ɛ (x, 0 = ϕ ɛ (x, x R. (2 u ɛ (, t L 2 (R, ρ is conserved in time. If ρ L (R, ρ 0 and ϕ ɛ is uniformly bounded in L 2 (R, then ρ(x u ɛ (x, t 2 is uniformly bounded in L (R, so that ρ( u ɛ (, t 2 µ, weakly in M(R. The ρ-weighted Wigner transform of u ɛ : w ɛ (x, t, ξ = ρ(xu ɛ (x + ɛz/2, tu ɛ (x ɛz/2, t exp( iξz dz. (3 2π R f ɛ (x, t, z := (F ξ z w ɛ (x, t, z - the Fourier transform of w ɛ in ξ. f ɛ satisfies the following equation: ft ɛ (x, t, z = ( 2 ρ(x + ɛz/2 + ρ(x ɛz/2 ρ (x ( ρ(x 2 ρ(x + ɛz/2 + ρ(x ɛz/2 + ( ɛ 2 ρ(x + ɛz/2 fz ɛ (x, t, z. ρ(x ɛz/2 f ɛ x (x, t, z (4 f ɛ (x, t, z Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 6 /
K ɛ (x, ξ := 2π R K2 ɛ (x, ξ := 2π R ( 2 ( ɛz ρ(x + ɛz/2 + ρ(x ɛz/2 ρ(x + ɛz/2 ρ(x ɛz/2 The Wigner transform w ɛ (x, t, ξ satisfies the following equation: exp( iξz dz (5 exp( iξz dz. (6 w ɛ t (x, t, ξ = K ɛ (x, w ɛ x (x, t, ξ ρ (x ρ(x K ɛ (x, w ɛ (x, t, ξ (7 K2 ɛ (x, w ɛ (x, t, ξ K2 ɛ (x, ξw ξ ɛ (x, t, ξ. Formally, K ɛ (x, ξ /ρ(xδ 0(ξ and K ɛ 2 (x, ξ (/ρ (xδ 0 (ξ, so that w ɛ (x, t, ξ converges to a function w(x, t, ξ which verifies the equation: w t(x, t, ξ = ρ(x wx (x, t, ξ + ρ (x ρ 2 (x ξw ξ(x, t, ξ. (8 Characteristics verifying the Hamiltonian system: x (t = ρ(x(t and ξ (t = ρ (x(t ρ 2 (x(t ξ(t. Uniqueness for the Hamiltonian system iff ρ C, (R. Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 7 /
Gérard, Markowich, Mauser, Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 997. Lions P.-L., Paul, Sur les mesures de Wigner, Rev. Matemática Iberoamericana, 993. 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, (R for uniqueness. FINITE DIFFERENCE APPROXIMATION of the transport equation f ɛ (x, t, z verifies the equation: ρ(xu ɛ t (x, t uɛ (x + ɛ, t u ɛ (x ɛ, t 2ɛ = 0, x R, t > 0. (9 f ɛ t (x, t, z = ρ(x f ɛ (x + ɛ/2, t, z + 2ɛρ(x + ɛz/2ρ(x + ɛ/2 ρ(x f ɛ (x ɛ/2, t, z 2ɛρ(x + ɛz/2ρ(x ɛ/2 f ɛ (x + ɛ/2, t, z + ρ(x 2ɛρ(x ɛz/2ρ(x + ɛ/2 ρ(x f ɛ (x ɛ/2, t, z + 2ɛρ(x ɛz/2ρ(x ɛ/2. (0 Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 8 /
wt ɛ (x, t, ξ = = ρ(x ( 4 ρ(x + ɛ/2 + ρ(x ɛ/2 + ρ(x ( 8 ρ(x + ɛ/2 + ρ(x ɛ/2 2 cos(ξ ( w ɛ (x + ɛ/2, t, ξ + w ɛ (x ɛ/2, t, ξ + ρ(x ( 4ɛ ρ(x + ɛ/2 ( + ɛ ρ(x 8ɛ ρ(x ɛ/2 ρ(x + ɛ/2 ρ(x ɛ/2 2 cos(ξ ( w ɛ (x + ɛ/2, t, ξ w ɛ (x ɛ/2, t, ξ. The limit equation is K ɛ (x, 2 cos(ξ w ɛ (x + ɛ/2, t, ξ w ɛ (x ɛ/2, t, ξ ɛ ( K 2 ɛ (x, 2 sin(ξ ( wξ ɛ (x + ɛ/2, t, ξ + w ξ ɛ (x ɛ/2, t, ξ K ɛ (x, 2 cos(ξ(w ɛ (x + ɛ/2, t, ξ + w ɛ (x ɛ/2, t, ξ ( K 2 ɛ (x, 2 sin(ξ ( wξ ɛ (x + ɛ/2, t, ξ w ξ ɛ (x ɛ/2, t, ξ w t(x, t, ξ = ρ(x cos(ξwx (x, t, ξ + ρ (x ρ 2 (x sin(ξw ξ(x, t, ξ. For the transport equation on a non-uniform grid y = g(x, we have w t(x, t, ξ = ( ρ(g(xg cos(ξwx (x, t, ξ + (x ρ(g(xg sin(ξw ξ (x, t, ξ. (x Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 9 /
Numerical simulations u 0 (y = ϕ γ ξ 0 (y y 0 exp(iξ 0 y 0 Legend: (e h = /200, uniform grid of size h for y (0, /2 and h/2 for y (/2, and y 0 = /4 (f-g h = /200, y = tan(πx/4, y 0 = /4; (h h = /200, y = sin(πx/3 for x (0, /2, y = sin(π( x/3 for x (/2, and y 0 = /2; (i h = /00, uniform grid of size h/8 and h/4 for y (0, /4 and y (3/4,, y = /4 + tan(x/4/2 for y (/4, 3/4, and y 0 = 7/8 We illustrate some phenomena, most of them being pathological and requiring further analysis: reflection-transmission problem at the interface between two piecewise uniform discrete media (see Fig. 4(e torsion of the rays of Geometric Optics, reflecting before touching the boundary of the domain (see Fig. 4(f-i (e ξ 0 = π/2h (f ξ 0 = π/2h min (g ξ 0 = π/h min (h ξ 0 = π/2h min (i ξ 0 = π/2h min Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 0 /
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, (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 Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 /
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, (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 Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 /