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Lecture 3

Lecture 3. Fourier Transform. f(t) – Function F( w ) – Fourier harmonic Time / Frequency exp(i k r) Co-ordinate / Wave-number Temporal (Spatial) spectrum: Spectral Analysis F = F( w ) : Spectral Distribution, Spectrum. Sunspot Data. Time Series Spectrum. FT: Properties.

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Lecture 3

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  1. Lecture 3

  2. Fourier Transform f(t) – Function F(w) – Fourier harmonic Time / Frequency exp(i k r) Co-ordinate / Wave-number Temporal (Spatial) spectrum: Spectral Analysis F = F(w) : Spectral Distribution, Spectrum

  3. Sunspot Data Time Series Spectrum

  4. FT: Properties - Linear superposition - Convolution • Correlation

  5. Discrete Fourier Transform Continuous: (-¥ … ¥ ) Discrete: ( h1 … hN ) k = 0 … (N-1) Discrete Spectrum

  6. DFT constraints Length-Scales Domain Size: L = X1-XN Step Size: DX = Xk – Xk-1 Wave-Numbers: Maximal wave-number: K = 2p/ DX Minimal wave-number: DK = 2p/ L Number of Fourier Harmonics: N (sampling rate, resolution)

  7. Sampling Discretization: Sample continuous function Nyquist critical frequency: wc = 1 / 2D w < wc

  8. Sampling

  9. Aliasing

  10. Aliasing

  11. Aliasing Kx -> -Kx

  12. Number of Harmonics N = 1…9 Density of the spectrum

  13. Fast Fourier Transform DFT: O(N2) calculation process FFT Algorithm: N = 2m O(N log2N) calculation process (Cooley, Turkey 1960, … Gauss 1805) N = 106 FFT : 30sec FT : 2 week

  14. PDE: Spectral Method PDE: Fourier decomposition in SPACE ODE solver a = a(k,t) Inverse Fourier transform A = A(x,t)

  15. Spectral Simulations Discrteize PDE: Sampling DFT: mostly FFT • Transform PDE to spectral ODE • Solve ODE (e.g., R-K) • Inverse transform to reconstruct solutions

  16. Spectral Method: Features Initial Value Problem • Calculate initial values in k-space Boundary Value Problem • Integrate boundaries into k-space Spatial Inhomogeneities • Introduce numerical variables to homogenize • Integrate during reconstruction

  17. Spectral Method: Problems • Shocks Discontinuity: D-> 0 Kcr = 1/ 2D->¥ Kmax <Kcr 2. Complex Boundaries Ill-known numerical instabilities; 3. Nonlinearities

  18. Spectral Method: Variants • Galerkin Method • Tau Method • Pseudo-spectral Method

  19. Comparison Spectral Method: Linear combination of continuous functions; Global approach; Finite Difference, Flux conservation: Array of piecewise functions; Local approach;

  20. + / - + (very) fast for smooth solutions + Exponential convergence + Best for turbulent spectrum • Shocks • Inhomogeneities • Complex Boundaries • Need for serial reconstruction (integration)

  21. Chaotic flows Post-processing: Partial Reconstruction

  22. end

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