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Lecture 3

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Fourier Transformf(t) – Function

F() – Fourier harmonic

Time / Frequency

exp(i k r)

Co-ordinate / Wave-number

Temporal (Spatial) spectrum: Spectral Analysis

F = F() : Spectral Distribution, Spectrum

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Sunspot Data

Time Series

Spectrum

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FT: Properties

- Linear superposition

- Convolution

- Correlation

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Discrete Fourier Transform

Continuous: (- … )

Discrete: ( h1 … hN )

k = 0 … (N-1)

Discrete Spectrum

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DFT constraints

Length-Scales

Domain Size: L = X1-XN

Step Size: X = Xk – Xk-1

Wave-Numbers:Maximal wave-number: K = 2/ XMinimal wave-number: K = 2/ L

Number of Fourier Harmonics: N(sampling rate, resolution)

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Sampling

Discretization:

Sample continuous

function

Nyquist critical frequency:

c = 1 / 2

c

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Sampling

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Aliasing

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Aliasing

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Aliasing

Kx -> -Kx

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Number of Harmonics

N = 1…9

Density of the spectrum

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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

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PDE: Spectral MethodPDE:

Fourier decomposition in SPACE

ODE solvera = a(k,t)

Inverse Fourier transformA = A(x,t)

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Spectral Simulations

Discrteize PDE: Sampling

DFT: mostly FFT

1. Transform PDE to spectral ODE

2. Solve ODE (e.g., R-K)

3. Inverse transform to reconstruct solutions

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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

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Spectral Method: Problems

1. ShocksDiscontinuity: 0

Kcr = 1/ 2 Kmax < Kcr

2. Complex BoundariesIll-known numerical instabilities;

3. Nonlinearities

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Spectral Method: Variants

• Galerkin Method

• Tau Method

• Pseudo-spectral Method

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Comparison

Spectral Method:

Linear combination of continuous functions;

Global approach;

Finite Difference, Flux conservation:

Array of piecewise functions;

Local approach;

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+ / -

+ (very) fast for smooth solutions

+ Exponential convergence

+ Best for turbulent spectrum

- Shocks

- Inhomogeneities

- Complex Boundaries

- Need for serial reconstruction (integration)

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Chaotic flows

Post-processing:

Partial Reconstruction

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