Computing Spectra of Linear Operators Using Finite Differences J. Nathan Kutz Department of Applied Mathematics University of Washington Seattle, WA 98195-2420 Email: (kutz@amath.washington.edu) Stability Workshop, Seattle, September 6-8, 2006
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)
Associated Eigenvalue Problem UW Applied Mathematics Separation of variables Eigenvalue problem Linear stability Re(λ) 0 Re(λ) > 0 spectrally stable spectrally unstable
Finite Differences and Taylor Series UW Applied Mathematics Taylor expand Add approximation error slope formula with error
Higher-Order Accuracy UW Applied Mathematics Taylor expand again 8 x (first) and subtract approximation error slope formula with improved error
Finite Difference Tables neighboring points determine accuracy
Forward and Backward Differences UW Applied Mathematics Required for incorporating boundary conditions asymmetric neighboring points
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
Boundary Conditions: Pinned pinned boundaries tri-diagonal matrix structure
Boundary Conditions: Periodic periodic boundaries tri-diagonal matrix structure with corners
Boundary Conditions: No Flux UW Applied Mathematics no flux condition no longer symmetry matrix
General Boundaries UW Applied Mathematics General (Sturm-Liouville) boundary conditions Difficult to incorporate into matrix structure shooting methods relaxation methods no longer symmetry matrix
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
Example: Mathieu Equation UW Applied Mathematics Classic Example Operator is self-adjoint (real spectrum) compute with matlab, maple, mathematica, or homemade code
Spectrum for Mathieu Equation UW Applied Mathematics q a a is eigenvalue
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 - 2.5 min Floquet theory
Calculating the Bands: Domain Length Increase the domain length (q=2) doubling gives 8x computational increase traditional way: very costly for recovering bands
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
Implementing Floquet Theory Floquet theory modifies matrix corners with Floquet slices
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
Example: Periodic NLS UW Applied Mathematics Consider the system with and Jacobi sine function
Spectrum of Periodic NLS k=0 k=0.7 k=0.9 k=1
Importance of Floquet Slicing µ = 0 µ 0
Accuracy and Convergence
Example: 2D Mathieu Equation Consider with Operation Count: O((N 2 ) 3 )=O(N 6 )
Laplacian in 2D Consider Discretize: Let must stack 2D data: periodic boundaries add structure
Laplacian in 2D nx=ny=4
Laplacian in 2D nx=ny=4 Matlab easily builds 2D Laplacian
Band Gap Structure A=-0.3 A=0 First three band-gap structures A=1 Quasi-momentum representation
2D Dominant Eigenfunctions First three eigenfunctions for µ x = µ y =0
2D Eigenfunctions First three eigenfunctions for µ x = µ y =1/4
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)