Anton ARNOLD. with N. Ben Abdallah (Toulouse), J. Geier (Vienna), C. Negulescu (Marseille) TU Vienna Institute for Analysis and Scientific Computing

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1 TECHNISCHE UNIVERSITÄT WIEN Asymptotically correct finite difference schemes for highly oscillatory ODEs Anton ARNOLD with N. Ben Abdallah (Toulouse, J. Geier (Vienna, C. Negulescu (Marseille TU Vienna Institute for Analysis and Scientific Computing UCLA, March 29 Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 1 / 21

2 Application: electron injection in semiconductor E often: coupled problem for all energies E > ; sharp transmission peaks w.r.t. E V(x future: couple oscillatory to evanescent regime: E < V(x stationary Schrödinger equation (1d: x 2 ϕ xx (x + ( E V(x ϕ(x =, x (, 1 }{{} 2m }{{} = ε 2 = a(x α> inhomogeneous open BCs: εϕ x ( + i a(ϕ( = 2i a(, εϕ x (1 i a(1ϕ(1 = reformulate as (backward IVP for ψ: ϕ(x = 2i a( εψ x(+i a(ψ( ψ(x Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 2 / 21

3 wavelength = O(ε/ a(x; GOAL: use stepsize h > ε accurate scheme that does NOT NEED to resolve the oscillations Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 3 / 21

4 GOAL: numerical scheme for ε 2 ϕ xx + a(xϕ = Outline: 1 analytic WKB-transformation of scalar ODE separate highly oscillatory term & smooth perturbation 2 scheme: correct uniformly in ε 3 approximation of oscillatory integrals 4 error estimates 5 numerical example 6 extension to vector systems Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 4 / 21

5 (1 WKB-method for analytic preprocessing of ODE WKB-ansatz for ε 2 ϕ xx + a(xϕ = : ϕ(x exp 1 ε ε p φ p (x p= zeroth order: first order: ϕ(x C exp ( ± i x ε a dτ ϕ(x C exp ( ± i x ε a dτ 4 a(x second order: ( exp ± i x ε a(τ ε 2 β(τdτ ϕ(x C, β := a 4 a(x 8a 3/2 5(a 2 32a 5/2 all: asymptotically correct for ε Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 5 / 21

6 WKB-FEM [Ben Abdallah-Pinaud, 26], [Negulescu, 28]: 1 st order WKB-approximation on (x n, x n+1 : ψ(x = phase: φ n (x = 1 ( A n e i φn(x ε a(x 4 x x n a(τ dτ, + B n e φn(x i ε real WKB-basis functions on cell (x n, x n+1 : α n (x = sin φ n+1(x ε 4 a(x n sin φn(x n+1 a(x, β n(x = ε sin φn(x ε sin φn(x n+1 ε need no-resonance condition for linear independence: kπ γ >, k Z φn(x n+1 ε use finite differences instead 4 a(x n+1 a(x Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 6 / 21

7 2 nd order WKB transformation [AA-B.Abdallah-Negulescu] 1 vector system: ( 4 aϕ(x U(x := ε( 4 aϕ (x a(x U = [ ( 1 a ε a ( ] + ε U 2β(x Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 7 / 21

8 2 nd order WKB transformation [AA-B.Abdallah-Negulescu] 1 vector system: ( 4 aϕ(x U(x := ε( 4 aϕ (x a(x Y (x := 1 ( i i U = 2 diagonalize dominant part: U Y = [ ( 1 a ε a ( ] + ε U 2β(x [ ( i a ε 2 ( ] β β ε a + ε 2 + ε Y β β Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 7 / 21

9 2 nd order WKB transformation [AA-B.Abdallah-Negulescu] 1 vector system: ( 4 aϕ(x U(x := ε( 4 aϕ (x a(x Y (x := 1 ( i i U = 2 diagonalize dominant part: U Y = [ ( 1 a ε a ( ] + ε U 2β(x [ ( i a ε 2 ( ] β β ε a + ε 2 + ε Y β β 3 eliminate leading oscillation: ( ( e i ε φ(x Z(x := e i ε φ(x Y (x Z βe 2i ε φ(x = ε βe 2i ε φ(x Z }{{} O(ε φ(x := x a ε 2 β dτ... phase of 2 nd order WKB-approximation Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 7 / 21

10 dominant oscillations eliminated: ex: ε 2 ϕ xx + a(xϕ = ; a(x = (x , x (, 1 Z much smoother than ϕ or U numerics easier epsi=1 1 epsi= x epsi= Psi.5 1 Z X X solution Rϕ(x for 2 values of ε; U L (,1 C, U L (,1 C ε ; solution Rz 1 (x: same frequency, amplitude = O(1 5 Z Z I Cε 2, Z Cε Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 8 / 21

11 (2 GOAL: ε uniform scheme ( Z β(xe 2i ε φ(x = ε β(xe 2i ε φ(x Z, }{{} =:N(x=O(1 x (, 1;Z( = Z I strong asymptotic limit Z ε= (x = Z I... trivial to capture numerically asymptotically correct scheme; i.e. error = O(ε p for some p 2 ε uniform approximation (truncated Picard iteration: Z(x = Z I +ε x x N(y 1 dy 1 Z I +ε 2 N(y 1 y1 N(y 2 dy 2 dy 1 Z I +O(ε 3 x 2 min(ε, x Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 9 / 21

12 (2 GOAL: ε uniform scheme ( Z β(xe 2i ε φ(x = ε β(xe 2i ε φ(x Z, }{{} =:N(x=O(1 x (, 1;Z( = Z I strong asymptotic limit Z ε= (x = Z I... trivial to capture numerically asymptotically correct scheme; i.e. error = O(ε p for some p 2 ε uniform approximation (truncated Picard iteration: Z(x = Z I +ε x x N(y 1 dy 1 Z I +ε 2 N(y 1 y1 N(y 2 dy 2 dy 1 Z I +O(ε 3 x 2 min(ε, x Remark: lower order WKB-transformations non-const. limit Z ε= only ε uniform scheme; i.e. error = O(1 [Lorenz-Jahnke-Lubich, 25] Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 9 / 21

13 (3 Approximation of oscillatory integrals GOAL: ε uniform approximation of J = (+ iterated s + to arbitrary h order! xn+1 x n β(y e 2i ε }{{}}{{ φ(y } dy smoothoscillatory asymptotic method [Iserles-Nørsett, 25]: [ ] β xn+1 2i J = iε ε 2φ e φ + O(min(ε 2, εh x n integration by parts yields higher ε orders but no h consistency for ODE Filon method [Iserles-Nørsett, 25]: would need exact moments J xn+1 x n π(y e 2i ε φ(y dy }{{} polynomial Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 1 / 21

14 modified asympt. method [AA-B.Abdallah-Negulescu 9] J = e 2i ε φ(xn xn+1 x n β(ye 2i xn+1 = iεe 2i ε φ(xn β d 2φ x n = iεe 2i ε φ(xn β 2φ (x n+1 ε [φ(y φ(xn] dy dy (e 2i ε [φ(y φ(xn] 1 dy (e 2i ε [φ(x n+1 φ(x n] 1 + O(min(εh, h 2 idea: include a zero of oscillatory factor by shift change ε power to h power 1 st order consistent, ε asymptotically correct ODE-method Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 11 / 21

15 Resulting methods first h-order: Z n+1 = (I + A 1 nz n second h-order: Z n+1 = (I + A 1 n + A 2 nz n ( A 1 n := iε 2 β 2φ (x n+1 A 2 n :=... e 2i ε φ(x n+1 e 2i e 2i ε φ(xn e 2i ε φ(x n+1 ε φ(xn Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 12 / 21

16 (4 Error estimates Theorem ([ABN 9] Z(x n Z n Cε p h α 1 min(ε, h, 1 n N, U(x n U n C hγ ε + Cεp h α 1 min(ε, h, 1 n N p = 1 α = 1, 2... h order of method γ... order of quadrature rule for phase φ(x = x a ε 2 β dτ Simpson (γ = 4 constraint h = O( ε for 2 nd order scheme φ exact for potential a(x piecewise linear (e.g. in RTD ε asymptotically correct scheme also for U simple improvement: p = 2 (note: Z(x Z I Cε 2 Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 13 / 21

17 (5 Numerical example ε 2 ϕ xx + a(xϕ = with a(x = (x phase φ = x a(τ ε 2 β(τdτ explicitly integrable eps=1 1 eps=1 2 eps=1 3 eps=1 4 eps= eps=1 1 eps=1 2 eps=1 3 eps=1 4 eps=1 5 Z num Z ref Z num Z ref h h L 1 (, 1 error of Z (as function of h for first & second order schemes schemes are asymptotically correct (error = O(ε 2, for h fixed! Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 14 / 21

18 Current conservation continuous current: j(x := εi(ϕ(xϕ (x = 1 ( 1 2 Z(x 1 discrete current: j n := 1 ( 2 Z n 1 Z 1 n slightly drifts: j n+1 j n =: ε 4 λ 2 j n Z(x = const in x x h=2 2 h=2 8 h=2 15 eps=1 1 2 j(x j( x simple fix by scaling possible: Z n+1 = ( 1 ε 4 λ 2 1/2 (I + A 1 n Z n Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 15 / 21

19 (6 Vector valued ODEs inital value problem: ϕ(x C d ϕ (x + 1 ε 2 A(xϕ(x =, ϕ( = ϕ, ϕ ( = ϕ. assumptions: R d d A(x = Q(xa(xQ (x > Q(x... orthogonal, smooth eigenvalues aj remain separated: a k (x a l (x δ >, a k (x 1 2 δ, k l a(x... diagonal matrix, smooth Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 16 / 21

20 analytic transformation I: th order WKB [LJL] 1 vector system: ( ( ( ϕ U(x := εa C 2d U 1 A 1 2 = 1 2ϕ ε A 1 2 A 2(A U }{{} only O(1 Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 17 / 21

21 analytic transformation I: th order WKB [LJL] 1 vector system: ( ( ( ϕ U(x := εa C 2d U 1 A 1 2 = 1 2ϕ ε A 1 2 A 2(A U }{{} only O(1 2 diagonalize dominant part: Y (x := 1 ( Q (x iq (x 2 iq (x Q (x [ ( Y i a 1 2 = ε a 1 2 µ := 1 2 Q A 1 2 (A 1 2 Q (... Kronecker product U ( 1 i i 1 µ + I ( Q Q ] Y, Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 17 / 21

22 analytic transformation II: th order WKB [LJL] 3 eliminate leading oscillation: ( e i ε φ(x Z(x := e i ε φ(x Y (x Z = 1 ( a 1 a 4 a 1 a +N ε (x Z }{{} =O(1 φ(x := x a 1 2 (τdτ... phase of th /1 st order WKB-approx., diag. matrix Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 18 / 21

23 analytic transformation II: th order WKB [LJL] 3 eliminate leading oscillation: ( e i ε φ(x Z(x := e i ε φ(x Y (x Z = 1 ( a 1 a 4 a 1 a +N ε (x Z }{{} =O(1 φ(x := x a 1 2 (τdτ... phase of th /1 st order WKB-approx., diag. matrix 4 optional improvement 1 st order WKB: ( a 1 4(xe Z(x i ε φ(x := a 1 4(xe i ε φ(x Y (x Z = Ñ ε (x Z }{{} off diag., O(1 Z = Z I + O(ε Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 18 / 21

24 2 approaches for vector systems [LJL 5]: th order WKB; standard treatment of oscillatory integrals [Geier-AA 9]: 1 st order WKB; for oscillatory integrals: shifted asymptotic method or moment-free Filon-type [Olver 6] both second h order Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 19 / 21

25 Numerical example example from [LJL 25]: d = 2, x [ 1, 1] ( ( a 1 2 (x = 3 2 x ( cos ξ(x sinξ(x Q(x = sin ξ(x cos ξ(x ξ(x = + x ( 1 1, with π arctan ( x 2, φ(x exactly computable [LJL 5]: error =O(h 2 (uniformly in ε, but NO ε-convergence of Z Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 2 / 21

26 rel. L 1 Fehler in η ε = 1.e 2 ε = 1.e 3 ε = 1.e 4 ε = 1.e 5 ε = 1.e 6 ε = 1.e 7 [LJL 5] [Geier-AA] h error in Z = O(εh 2 Anton ARNOLD (TU Vienna Asymptotically correct FD schemes 21 / 21

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