Computing Spectra of Linear Operators Using Finite Differences

Transcription

1 Computing Spectra of Linear Operators Using Finite Differences J. Nathan Kutz Department of Applied Mathematics University of Washington Seattle, WA Stability Workshop, Seattle, September 6-8, 2006

2 Introduction: Spectral Stability UW Applied Mathematics Consider the nonlinear PDE evolution Equilibrium solution Linear stability perturbation B. Deconinck and J. N. Kutz, J. Comp. Phys. (2006)

3 Associated Eigenvalue Problem UW Applied Mathematics Separation of variables Eigenvalue problem Linear stability Re(λ) 0 Re(λ) > 0 spectrally stable spectrally unstable

4 Finite Differences and Taylor Series UW Applied Mathematics Taylor expand Add approximation error slope formula with error

5 Higher-Order Accuracy UW Applied Mathematics Taylor expand again 8 x (first) and subtract approximation error slope formula with improved error

6 Finite Difference Tables neighboring points determine accuracy

7 Forward and Backward Differences UW Applied Mathematics Required for incorporating boundary conditions asymmetric neighboring points

8 Numerical Round-Off UW Applied Mathematics Consider the error in approximating the first derivative The error includes round-off and truncation round-off Assume round-off and minimum at truncation round-off error dominates below

9 Boundary Conditions: Pinned pinned boundaries tri-diagonal matrix structure

10 Boundary Conditions: Periodic periodic boundaries tri-diagonal matrix structure with corners

11 Boundary Conditions: No Flux UW Applied Mathematics no flux condition no longer symmetry matrix

12 General Boundaries UW Applied Mathematics General (Sturm-Liouville) boundary conditions Difficult to incorporate into matrix structure shooting methods relaxation methods no longer symmetry matrix

13 Algorithm UW Applied Mathematics choose domain length and discretization size construct linear operator implement boundary conditions use eigenvalue/eigenvector solver: O(N 3 ) (or shooting/relaxation methods) construct eigenfunctions Easily extends to vectors and higher dimensions

14 Example: Mathieu Equation UW Applied Mathematics Classic Example Operator is self-adjoint (real spectrum) compute with matlab, maple, mathematica, or homemade code

15 Spectrum for Mathieu Equation UW Applied Mathematics q a a is eigenvalue

16 Computing the Ground State UW Applied Mathematics Convergence study and CPU time (q=2) 0.43 sec 0.01 sec What about band-gap structure 1 hour 0.77 sec increase domain length beyond Matlab7 s ability min Floquet theory

17 Calculating the Bands: Domain Length Increase the domain length (q=2) doubling gives 8x computational increase traditional way: very costly for recovering bands

18 Calculating the Bands: Floquet Theory Make use of Floquet (Bloch) theory with Floquet (characteristic) exponents keep fixed domain discretize larger period solutions solve D O(N 3 ) equations

19 Implementing Floquet Theory Floquet theory modifies matrix corners with Floquet slices

20 Floquet Theory vs. Domain Length Compare methods for computing band (q=2) band density 9 min 8 sec Use Floquet Theory! - 1 min beyond Matlab7 s ability - 10 min

21 Example: Periodic NLS UW Applied Mathematics Consider the system with and Jacobi sine function

22 Spectrum of Periodic NLS k=0 k=0.7 k=0.9 k=1

23 Importance of Floquet Slicing µ = 0 µ 0

24 Accuracy and Convergence

25 Example: 2D Mathieu Equation Consider with Operation Count: O((N 2 ) 3 )=O(N 6 )

26 Laplacian in 2D Consider Discretize: Let must stack 2D data: periodic boundaries add structure

27 Laplacian in 2D nx=ny=4

28 Laplacian in 2D nx=ny=4 Matlab easily builds 2D Laplacian

29 Band Gap Structure A=-0.3 A=0 First three band-gap structures A=1 Quasi-momentum representation

30 2D Dominant Eigenfunctions First three eigenfunctions for µ x = µ y =0

31 2D Eigenfunctions First three eigenfunctions for µ x = µ y =1/4

32 Summary and Conclusions simple, simple, simple boundary conditions at edge of matrices eigenvalue solvers make use of sparse structure Floquet theory for resolution of bands costly/impractical for 2D-3D problems B. Deconinck and J. N. Kutz, J. Comp. Phys. (2006)