Anton ARNOLD. with N. Ben Abdallah (Toulouse), J. Geier (Vienna), C. Negulescu (Marseille) TU Vienna Institute for Analysis and Scientific Computing
|
|
- Alan Cooper
- 5 years ago
- Views:
Transcription
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
WKB-based schemes for two-band Schrödinger equations in the highly ascillatory regime
ASC Report No. 25/2011 WKB-based schemes for two-band Schrödinger equations in the highly ascillatory regime Jens Geier, Anton Arnold Institute for Analysis and Scientific Computing Vienna University of
More informationNumerical solution of the Schrödinger equation on unbounded domains
Numerical solution of the Schrödinger equation on unbounded domains Maike Schulte Institute for Numerical and Applied Mathematics Westfälische Wilhelms-Universität Münster Cetraro, September 6 OUTLINE
More informationPartial differential equation for temperature u(x, t) in a heat conducting insulated rod along the x-axis is given by the Heat equation:
Chapter 7 Heat Equation Partial differential equation for temperature u(x, t) in a heat conducting insulated rod along the x-axis is given by the Heat equation: u t = ku x x, x, t > (7.1) Here k is a constant
More informationCalculation of Highly Oscillatory Integrals by Quadrature Method
Calculation of Highly Oscillatory Integrals by Quadrature Methods Krishna Thapa Texas A&M University, College Station, TX May 18, 211 Outline Purpose Why Filon Type Quadrature Methods Our Expectations
More informationA new multiscale discontinuous Galerkin method for the one-dimensional stationary Schrödinger equation
A new multiscale discontinuous Galerkin method for the one-dimensional stationary Schrödinger equation Bo Dong, Chi-Wang Shu and Wei Wang Abstract In this paper, we develop and analyze a new multiscale
More informationA Numerical Approach To The Steepest Descent Method
A Numerical Approach To The Steepest Descent Method K.U. Leuven, Division of Numerical and Applied Mathematics Isaac Newton Institute, Cambridge, Feb 15, 27 Outline 1 One-dimensional oscillatory integrals
More informationThe WKB local discontinuous Galerkin method for the simulation of Schrödinger equation in a resonant tunneling diode
The WKB local discontinuous Galerkin method for the simulation of Schrödinger equation in a resonant tunneling diode Wei Wang and Chi-Wang Shu Division of Applied Mathematics Brown University Providence,
More informationSecond Order ODEs. CSCC51H- Numerical Approx, Int and ODEs p.130/177
Second Order ODEs Often physical or biological systems are best described by second or higher-order ODEs. That is, second or higher order derivatives appear in the mathematical model of the system. For
More informationHigh Frequency Scattering by Convex Polygons Stephen Langdon
Bath, October 28th 2005 1 High Frequency Scattering by Convex Polygons Stephen Langdon University of Reading, UK Joint work with: Simon Chandler-Wilde Steve Arden Funded by: Leverhulme Trust University
More informationNonconstant Coefficients
Chapter 7 Nonconstant Coefficients We return to second-order linear ODEs, but with nonconstant coefficients. That is, we consider (7.1) y + p(t)y + q(t)y = 0, with not both p(t) and q(t) constant. The
More informationPreliminary Examination, Numerical Analysis, August 2016
Preliminary Examination, Numerical Analysis, August 2016 Instructions: This exam is closed books and notes. The time allowed is three hours and you need to work on any three out of questions 1-4 and any
More informationChemistry 532 Problem Set 7 Spring 2012 Solutions
Chemistry 53 Problem Set 7 Spring 01 Solutions 1. The study of the time-independent Schrödinger equation for a one-dimensional particle subject to the potential function leads to the differential equation
More informationOrthogonality of hat functions in Sobolev spaces
1 Orthogonality of hat functions in Sobolev spaces Ulrich Reif Technische Universität Darmstadt A Strobl, September 18, 27 2 3 Outline: Recap: quasi interpolation Recap: orthogonality of uniform B-splines
More informationFinite Difference Methods for Boundary Value Problems
Finite Difference Methods for Boundary Value Problems October 2, 2013 () Finite Differences October 2, 2013 1 / 52 Goals Learn steps to approximate BVPs using the Finite Difference Method Start with two-point
More informationGaussian Beam Approximations
Gaussian Beam Approximations Olof Runborg CSC, KTH Joint with Hailiang Liu, Iowa, and Nick Tanushev, Austin Technischen Universität München München, January 2011 Olof Runborg (KTH) Gaussian Beam Approximations
More informationFinal Exam May 4, 2016
1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.
More informationMB4018 Differential equations
MB4018 Differential equations Part II http://www.staff.ul.ie/natalia/mb4018.html Prof. Natalia Kopteva Spring 2015 MB4018 (Spring 2015) Differential equations Part II 0 / 69 Section 1 Second-Order Linear
More informationEscape Problems & Stochastic Resonance
Escape Problems & Stochastic Resonance 18.S995 - L08 & 09 dunkel@mit.edu Examples Illustration by J.P. Cartailler. Copyright 2007, Symmation LLC. transport & evolution: stochastic escape problems http://evolutionarysystemsbiology.org/intro/
More informationFirst-Order ODE: Separable Equations, Exact Equations and Integrating Factor
First-Order ODE: Separable Equations, Exact Equations and Integrating Factor Department of Mathematics IIT Guwahati REMARK: In the last theorem of the previous lecture, you can change the open interval
More informationOn second order sufficient optimality conditions for quasilinear elliptic boundary control problems
On second order sufficient optimality conditions for quasilinear elliptic boundary control problems Vili Dhamo Technische Universität Berlin Joint work with Eduardo Casas Workshop on PDE Constrained Optimization
More informationUniformly accurate averaging numerical schemes for oscillatory evolution equations
Uniformly accurate averaging numerical schemes for oscillatory evolution equations Philippe Chartier University of Rennes, INRIA Joint work with M. Lemou (University of Rennes-CNRS), F. Méhats (University
More informationExponential integrators
Exponential integrators Marlis Hochbruck Heinrich-Heine University Düsseldorf, Germany Alexander Ostermann University Innsbruck, Austria Helsinki, May 25 p.1 Outline Time dependent Schrödinger equations
More informationZ. Zhou On the classical limit of a time-dependent self-consistent field system: analysis. computation
On the classical limit of a time-dependent self-consistent field system: analysis and computation Zhennan Zhou 1 joint work with Prof. Shi Jin and Prof. Christof Sparber. 1 Department of Mathematics Duke
More informationOn the Stability of the Best Reply Map for Noncooperative Differential Games
On the Stability of the Best Reply Map for Noncooperative Differential Games Alberto Bressan and Zipeng Wang Department of Mathematics, Penn State University, University Park, PA, 68, USA DPMMS, University
More informationOR MSc Maths Revision Course
OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:30-12:30 Mathematics revision
More informationPeriodic oscillations in the Gross-Pitaevskii equation with a parabolic potential
Periodic oscillations in the Gross-Pitaevskii equation with a parabolic potential Dmitry Pelinovsky 1 and Panos Kevrekidis 2 1 Department of Mathematics, McMaster University, Hamilton, Ontario, Canada
More informationExponential multistep methods of Adams-type
Exponential multistep methods of Adams-type Marlis Hochbruck and Alexander Ostermann KARLSRUHE INSTITUTE OF TECHNOLOGY (KIT) 0 KIT University of the State of Baden-Wuerttemberg and National Laboratory
More informationTransmission across potential wells and barriers
3 Transmission across potential wells and barriers The physics of transmission and tunneling of waves and particles across different media has wide applications. In geometrical optics, certain phenomenon
More informationPage 404. Lecture 22: Simple Harmonic Oscillator: Energy Basis Date Given: 2008/11/19 Date Revised: 2008/11/19
Page 404 Lecture : Simple Harmonic Oscillator: Energy Basis Date Given: 008/11/19 Date Revised: 008/11/19 Coordinate Basis Section 6. The One-Dimensional Simple Harmonic Oscillator: Coordinate Basis Page
More informationA Survey of Computational High Frequency Wave Propagation II. Olof Runborg NADA, KTH
A Survey of Computational High Frequency Wave Propagation II Olof Runborg NADA, KTH High Frequency Wave Propagation CSCAMM, September 19-22, 2005 Numerical methods Direct methods Wave equation (time domain)
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More informationMAE294B/SIOC203B: Methods in Applied Mechanics Winter Quarter sgls/mae294b Solution IV
MAE9B/SIOC3B: Methods in Applied Mechanics Winter Quarter 8 http://webengucsdedu/ sgls/mae9b 8 Solution IV (i The equation becomes in T Applying standard WKB gives ɛ y TT ɛte T y T + y = φ T Te T φ T +
More informationHigh-Gain Observers in Nonlinear Feedback Control. Lecture # 2 Separation Principle
High-Gain Observers in Nonlinear Feedback Control Lecture # 2 Separation Principle High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 1/4 The Class of Systems ẋ = Ax + Bφ(x,
More informationMATH 819 FALL We considered solutions of this equation on the domain Ū, where
MATH 89 FALL. The D linear wave equation weak solutions We have considered the initial value problem for the wave equation in one space dimension: (a) (b) (c) u tt u xx = f(x, t) u(x, ) = g(x), u t (x,
More informationGroup Method. December 16, Oberwolfach workshop Dynamics of Patterns
CWI, Amsterdam heijster@cwi.nl December 6, 28 Oberwolfach workshop Dynamics of Patterns Joint work: A. Doelman (CWI/UvA), T.J. Kaper (BU), K. Promislow (MSU) Outline 2 3 4 Interactions of localized structures
More informationSummary of time independent electrodynamics
hapter 10 Summary of time independent electrodynamics 10.1 Electrostatics Physical law oulomb s law charges as origin of electric field Superposition principle ector of the electric field E(x) in vacuum
More information3.1 Interpolation and the Lagrange Polynomial
MATH 4073 Chapter 3 Interpolation and Polynomial Approximation Fall 2003 1 Consider a sample x x 0 x 1 x n y y 0 y 1 y n. Can we get a function out of discrete data above that gives a reasonable estimate
More informationThe Factorization Method for Inverse Scattering Problems Part I
The Factorization Method for Inverse Scattering Problems Part I Andreas Kirsch Madrid 2011 Department of Mathematics KIT University of the State of Baden-Württemberg and National Large-scale Research Center
More informationNumerical Analysis of Differential Equations Numerical Solution of Elliptic Boundary Value
Numerical Analysis of Differential Equations 188 5 Numerical Solution of Elliptic Boundary Value Problems 5 Numerical Solution of Elliptic Boundary Value Problems TU Bergakademie Freiberg, SS 2012 Numerical
More informationSuboptimal feedback control of PDEs by solving Hamilton-Jacobi Bellman equations on sparse grids
Suboptimal feedback control of PDEs by solving Hamilton-Jacobi Bellman equations on sparse grids Jochen Garcke joint work with Axel Kröner, INRIA Saclay and CMAP, Ecole Polytechnique Ilja Kalmykov, Universität
More informationA stroboscopic averaging technique for highly-oscillatory Schrödinger equation
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,
More informationQuantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.
Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.
More information{ } is an asymptotic sequence.
AMS B Perturbation Methods Lecture 3 Copyright by Hongyun Wang, UCSC Recap Iterative method for finding asymptotic series requirement on the iteration formula to make it work Singular perturbation use
More informationLECTURE 5: THE METHOD OF STATIONARY PHASE
LECTURE 5: THE METHOD OF STATIONARY PHASE Some notions.. A crash course on Fourier transform For j =,, n, j = x j. D j = i j. For any multi-index α = (α,, α n ) N n. α = α + + α n. α! = α! α n!. x α =
More informationCut-on, cut-off transition of sound in slowly varying flow ducts
Cut-on, cut-off transition of sound in slowly varying flow ducts Sjoerd W. Rienstra 19-3-1 Department of Mathematics and Computing Science, Eindhoven University of Technology, The Netherlands, s.w.rienstra@tue.nl
More informationQuantum Fokker-Planck models: kinetic & operator theory approaches
TECHNISCHE UNIVERSITÄT WIEN Quantum Fokker-Planck models: kinetic & operator theory approaches Anton ARNOLD with F. Fagnola (Milan), L. Neumann (Vienna) TU Vienna Institute for Analysis and Scientific
More information' Liberty and Umou Ono and Inseparablo "
3 5? #< q 8 2 / / ) 9 ) 2 ) > < _ / ] > ) 2 ) ) 5 > x > [ < > < ) > _ ] ]? <
More informationEach problem is worth 34 points. 1. Harmonic Oscillator Consider the Hamiltonian for a simple harmonic oscillator. 2ml 2 0. d 2
Physics 443 Prelim # with solutions March 7, 8 Each problem is worth 34 points.. Harmonic Oscillator Consider the Hamiltonian for a simple harmonic oscillator H p m + mω x (a Use dimensional analysis to
More informationGaussian Beam Approximations of High Frequency Waves
Gaussian Beam Approximations of High Frequency Waves Olof Runborg Department of Mathematics, KTH Joint with Hailiang Liu, Iowa, James Ralston, UCLA, Nick Tanushev, Z-Terra Corp. ESF Exploratory Workshop
More informationDynamics at the Horsetooth Volume 2A, Focused Issue: Asymptotics and Perturbations, 2010.
Dynamics at the Horsetooth Volume 2A, Focused Issue: Asymptotics Perturbations, 2010. Perturbation Theory the WKB Method Department of Mathematics Colorado State University shinn@math.colostate.edu Report
More informationFrom the Newton equation to the wave equation in some simple cases
From the ewton equation to the wave equation in some simple cases Xavier Blanc joint work with C. Le Bris (EPC) and P.-L. Lions (Collège de France) Université Paris Diderot, FRACE http://www.ann.jussieu.fr/
More informationChapter Two: Numerical Methods for Elliptic PDEs. 1 Finite Difference Methods for Elliptic PDEs
Chapter Two: Numerical Methods for Elliptic PDEs Finite Difference Methods for Elliptic PDEs.. Finite difference scheme. We consider a simple example u := subject to Dirichlet boundary conditions ( ) u
More information(f(x) P 3 (x)) dx. (a) The Lagrange formula for the error is given by
1. QUESTION (a) Given a nth degree Taylor polynomial P n (x) of a function f(x), expanded about x = x 0, write down the Lagrange formula for the truncation error, carefully defining all its elements. How
More informationIntroduction to PDEs and Numerical Methods Tutorial 11. 2D elliptic equations
Platzalter für Bild, Bild auf Titelfolie inter das Logo einsetzen Introduction to PDEs and Numerical Metods Tutorial 11. 2D elliptic equations Dr. Noemi Friedman, 3. 1. 215. Overview Introduction (classification
More information2.20 Fall 2018 Math Review
2.20 Fall 2018 Math Review September 10, 2018 These notes are to help you through the math used in this class. This is just a refresher, so if you never learned one of these topics you should look more
More informationNEW NUMERICAL INTEGRATORS BASED ON SOLVABILITY AND SPLITTING Fernando Casas Universitat Jaume I, Castellón, Spain Fernando.Casas@uji.es (on sabbatical leave at DAMTP, University of Cambridge) Edinburgh,
More informationDepartment of Applied Mathematics and Theoretical Physics. AMA 204 Numerical analysis. Exam Winter 2004
Department of Applied Mathematics and Theoretical Physics AMA 204 Numerical analysis Exam Winter 2004 The best six answers will be credited All questions carry equal marks Answer all parts of each question
More informationHomogenization of micro-resonances and localization of waves.
Homogenization of micro-resonances and localization of waves. Valery Smyshlyaev University College London, UK July 13, 2012 (joint work with Ilia Kamotski UCL, and Shane Cooper Bath/ Cardiff) Valery Smyshlyaev
More informationEnhanced resolution in structured media
Enhanced resolution in structured media Eric Bonnetier w/ H. Ammari (Ecole Polytechnique), and Yves Capdeboscq (Oxford) Outline : 1. An experiment of super resolution 2. Small volume asymptotics 3. Periodicity
More informationLate-time tails of self-gravitating waves
Late-time tails of self-gravitating waves (non-rigorous quantitative analysis) Piotr Bizoń Jagiellonian University, Kraków Based on joint work with Tadek Chmaj and Andrzej Rostworowski Outline: Motivation
More informationMath 4263 Homework Set 1
Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georgia Tech PHYS 612 Mathematical Methods of Physics I Instructor: Predrag Cvitanović Fall semester 2012 Homework Set #5 due October 2, 2012 == show all your work for maximum credit, == put labels, title,
More informationMATH 220: Problem Set 3 Solutions
MATH 220: Problem Set 3 Solutions Problem 1. Let ψ C() be given by: 0, x < 1, 1 + x, 1 < x < 0, ψ(x) = 1 x, 0 < x < 1, 0, x > 1, so that it verifies ψ 0, ψ(x) = 0 if x 1 and ψ(x)dx = 1. Consider (ψ j )
More informationInverse problems Total Variation Regularization Mark van Kraaij Casa seminar 23 May 2007 Technische Universiteit Eindh ove n University of Technology
Inverse problems Total Variation Regularization Mark van Kraaij Casa seminar 23 May 27 Introduction Fredholm first kind integral equation of convolution type in one space dimension: g(x) = 1 k(x x )f(x
More informationSeparation of Variables in Linear PDE: One-Dimensional Problems
Separation of Variables in Linear PDE: One-Dimensional Problems Now we apply the theory of Hilbert spaces to linear differential equations with partial derivatives (PDE). We start with a particular example,
More informationOn the quadrature of multivariate highly oscillatory integrals over non-polytope domains
On the quadrature of multivariate highly oscillatory integrals over non-polytope domains Sheehan Olver February 16, 2006 Abstract In this paper, we present a Levin-type method for approximating multivariate
More informationWave propagation in optical waveguides
Wave propagation in optical waveguides Giulio Ciraolo November, 005 Abstract We present a mathematical framework for studying the problem of electromagnetic wave propagation in a -D or 3-D optical waveguide
More informationArtificial boundary conditions for dispersive equations. Christophe Besse
Artificial boundary conditions for dispersive equations by Christophe Besse Institut Mathématique de Toulouse, Université Toulouse 3, CNRS Groupe de travail MathOcéan Bordeaux INSTITUT de MATHEMATIQUES
More informationLast time: Normalization of eigenfunctions which belong to continuously variable eigenvalues. often needed - alternate method via JWKB next lecture
1. identities. ψδk, ψδp, ψδe, ψbox 3. trick using box normalization 4. Last time: Normalization of eigenfunctions which belong to continuously variable eigenvalues. # states # particles δθ δx dn de L 1/
More informationDifferential Equations 2280 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 2015 at 12:50pm
Differential Equations 228 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 25 at 2:5pm Instructions: This in-class exam is 5 minutes. No calculators, notes, tables or books. No answer check is
More information(1 + 2y)y = x. ( x. The right-hand side is a standard integral, so in the end we have the implicit solution. y(x) + y 2 (x) = x2 2 +C.
Midterm 1 33B-1 015 October 1 Find the exact solution of the initial value problem. Indicate the interval of existence. y = x, y( 1) = 0. 1 + y Solution. We observe that the equation is separable, and
More informationMath 46, Applied Math (Spring 2008): Final
Math 46, Applied Math (Spring 2008): Final 3 hours, 80 points total, 9 questions, roughly in syllabus order (apart from short answers) 1. [16 points. Note part c, worth 7 points, is independent of the
More informationA Direct Method for reconstructing inclusions from Electrostatic Data
A Direct Method for reconstructing inclusions from Electrostatic Data Isaac Harris Texas A&M University, Department of Mathematics College Station, Texas 77843-3368 iharris@math.tamu.edu Joint work with:
More informationReconstruction Scheme for Active Thermography
Reconstruction Scheme for Active Thermography Gen Nakamura gnaka@math.sci.hokudai.ac.jp Department of Mathematics, Hokkaido University, Japan Newton Institute, Cambridge, Sept. 20, 2011 Contents.1.. Important
More informationEddy viscosity of cellular flows by upscaling
Eddy viscosity of cellular flows by upscaling Alexei Novikov a a California Institute of Technology, Applied & Computational Mathematics 1200 E. California Boulevard, MC 217-50, Pasadena, CA 91125, USA
More informationSome Remarks on the Reissner Mindlin Plate Model
Some Remarks on the Reissner Mindlin Plate Model Alexandre L. Madureira LNCC Brazil Talk at the Workshop of Numerical Analysis of PDEs LNCC February 10, 2003 1 Outline The 3D Problem and its Modeling Full
More informationLucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche
Scuola di Dottorato THE WAVE EQUATION Lucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche Lucio Demeio - DIISM wave equation 1 / 44 1 The Vibrating String Equation 2 Second
More informationAsymptotic Behavior of Waves in a Nonuniform Medium
Available at http://pvamuedu/aam Appl Appl Math ISSN: 1932-9466 Vol 12, Issue 1 June 217, pp 217 229 Applications Applied Mathematics: An International Journal AAM Asymptotic Behavior of Waves in a Nonuniform
More informationSection 3.4. Second Order Nonhomogeneous. The corresponding homogeneous equation. is called the reduced equation of (N).
Section 3.4. Second Order Nonhomogeneous Equations y + p(x)y + q(x)y = f(x) (N) The corresponding homogeneous equation y + p(x)y + q(x)y = 0 (H) is called the reduced equation of (N). 1 General Results
More information1 Separation of Variables
Jim ambers ENERGY 281 Spring Quarter 27-8 ecture 2 Notes 1 Separation of Variables In the previous lecture, we learned how to derive a PDE that describes fluid flow. Now, we will learn a number of analytical
More informationMidterm Solution
18303 Midterm Solution Problem 1: SLP with mixed boundary conditions Consider the following regular) Sturm-Liouville eigenvalue problem consisting in finding scalars λ and functions v : [0, b] R b > 0),
More informationA Time-Splitting Spectral Method for the Generalized Zakharov System in Multi-dimensions
A Time-Splitting Spectral Method for the Generalized Zakharov System in Multi-dimensions Shi Jin and Chunxiong Zheng Abstract The generalized Zakharov system couples a dispersive field E (scalar or vectorial)
More informationIterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems Luca Bergamaschi e-mail: berga@dmsa.unipd.it - http://www.dmsa.unipd.it/ berga Department of Mathematical Methods and Models for Scientific Applications University
More informationNATIONAL UNIVERSITY OF SINGAPORE PC5215 NUMERICAL RECIPES WITH APPLICATIONS. (Semester I: AY ) Time Allowed: 2 Hours
NATIONAL UNIVERSITY OF SINGAPORE PC55 NUMERICAL RECIPES WITH APPLICATIONS (Semester I: AY 0-) Time Allowed: Hours INSTRUCTIONS TO CANDIDATES. This examination paper contains FIVE questions and comprises
More informationLECTURE 3: DISCRETE GRADIENT FLOWS
LECTURE 3: DISCRETE GRADIENT FLOWS Department of Mathematics and Institute for Physical Science and Technology University of Maryland, USA Tutorial: Numerical Methods for FBPs Free Boundary Problems and
More informationA A x i x j i j (i, j) (j, i) Let. Compute the value of for and
7.2 - Quadratic Forms quadratic form on is a function defined on whose value at a vector in can be computed by an expression of the form, where is an symmetric matrix. The matrix R n Q R n x R n Q(x) =
More informationSemi-Classical Dynamics Using Hagedorn Wavepackets
Semi-Classical Dynamics Using Hagedorn Wavepackets Leila Taghizadeh June 6, 2013 Leila Taghizadeh () Semi-Classical Dynamics Using Hagedorn Wavepackets June 6, 2013 1 / 56 Outline 1 The Schrödinger Equation
More informationEstimation of probability density functions by the Maximum Entropy Method
Estimation of probability density functions by the Maximum Entropy Method Claudio Bierig, Alexey Chernov Institute for Mathematics Carl von Ossietzky University Oldenburg alexey.chernov@uni-oldenburg.de
More information6 Non-homogeneous Heat Problems
6 Non-homogeneous Heat Problems Up to this point all the problems we have considered for the heat or wave equation we what we call homogeneous problems. This means that for an interval < x < l the problems
More informationLu u. µ (, ) the equation. has the non-zero solution
MODULE 18 Topics: Eigenvalues and eigenvectors for differential operators In analogy to the matrix eigenvalue problem Ax = λx we shall consider the eigenvalue problem Lu = µu where µ is a real or complex
More informationFinite Element Methods
Solving Operator Equations Via Minimization We start with several definitions. Definition. Let V be an inner product space. A linear operator L: D V V is said to be positive definite if v, Lv > for every
More informationComputational High Frequency Wave Propagation
Computational High Frequency Wave Propagation Olof Runborg CSC, KTH Isaac Newton Institute Cambridge, February 2007 Olof Runborg (KTH) High-Frequency Waves INI, 2007 1 / 52 Outline 1 Introduction, background
More informationPhysics 215b: Problem Set 5
Physics 25b: Problem Set 5 Prof. Matthew Fisher Solutions prepared by: James Sully April 3, 203 Please let me know if you encounter any typos in the solutions. Problem 20 Let us write the wavefunction
More informationPH.D. PRELIMINARY EXAMINATION MATHEMATICS
UNIVERSITY OF CALIFORNIA, BERKELEY SPRING SEMESTER 207 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials NAME PH.D. PRELIMINARY EXAMINATION MATHEMATICS Problem
More informationyou expect to encounter difficulties when trying to solve A x = b? 4. A composite quadrature rule has error associated with it in the following form
Qualifying exam for numerical analysis (Spring 2017) Show your work for full credit. If you are unable to solve some part, attempt the subsequent parts. 1. Consider the following finite difference: f (0)
More informationOPERATORS WITH SINGULAR CONTINUOUS SPECTRUM, V. SPARSE POTENTIALS. B. Simon 1 and G. Stolz 2
OPERATORS WITH SINGULAR CONTINUOUS SPECTRUM, V. SPARSE POTENTIALS B. Simon 1 and G. Stolz 2 Abstract. By presenting simple theorems for the absence of positive eigenvalues for certain one-dimensional Schrödinger
More informationIntegration, differentiation, and root finding. Phys 420/580 Lecture 7
Integration, differentiation, and root finding Phys 420/580 Lecture 7 Numerical integration Compute an approximation to the definite integral I = b Find area under the curve in the interval Trapezoid Rule:
More informationLinear Hyperbolic Systems
Linear Hyperbolic Systems Professor Dr E F Toro Laboratory of Applied Mathematics University of Trento, Italy eleuterio.toro@unitn.it http://www.ing.unitn.it/toro October 8, 2014 1 / 56 We study some basic
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 15 Heat with a source So far we considered homogeneous wave and heat equations and the associated initial value problems on the whole line, as
More informationKAM for quasi-linear KdV
KAM for quasi-linear KdV Massimiliano Berti ST Etienne de Tinée, 06-02-2014 KdV t u + u xxx 3 x u 2 + N 4 (x, u, u x, u xx, u xxx ) = 0, x T Quasi-linear Hamiltonian perturbation N 4 := x {( u f )(x, u,
More information