Basic Equations

Calculating a derivative using the differentation theorem of the Fourier Transform is in the mathematical sense a convolution of the function $f(x)$ with $ik$, where $k$ is the wavenumber and $i$ the imaginary unit. This can also be formulated as a matrix-vector product involving so-called Toeplitz matrices. An elegant (but inefficient) way of performing a derivative operation on a space-dependent function described on the Chebyshev collocation points is by defining a derivative matrix $D_{ij}$

$$ D_{ij} \ = \ -\frac{2 N^2 + 1}{6} \hspace{1.5cm} \text{for i = j = N} $$$$ D_{ij} \ = \ -\frac{1}{2} \frac{x_i}{1-x_i^2} \hspace{1.5cm} \text{for i = j = 1,2,...,N-1} $$$$ D_{ij} \ = \ \frac{c_i}{c_j} \frac{(-1)^{i+j}}{x_i - x_j} \hspace{1.5cm} \text{for i $\neq$ j = 0,1,...,N}$$

where $N+1$ is the number of Chebyshev collocation points $ \ x_i = cos(i\pi / N)$, $ \ i=0,...,N$ and the $c_i$ are given as

$$ c_i = 2 \hspace{1.5cm} \text{for i = 0 or N} $$$$ c_i = 1 \hspace{1.5cm} \text{otherwise} $$

This differentiation matrix allows us to write the derivative of the function $f_i = f(x_i)$ (possibly depending on time) simply as

$$\partial_x u_i = D_{ij} \ u_j$$

where the right-hand side is a matrix-vector product, and the Einstein summation convention applies.



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# This is a configuration step for the exercise. Please run it before calculating the derivative!
import numpy as np
import matplotlib.pyplot as plt

# Show the plots in the Notebook.
plt.switch_backend("nbagg")

Exercise 1

Define a python function call "get_chebymatrix(nx)" that initializes the Chebyshev derivative matrix $D{ij}$



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# IMPLEMENT THE CHEBYSHEV DERIVATIVE MATRIX METHOD HERE!
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Exercise 2

Calculate the numerical derivative by applying the differentiation matrix $D_{ij}$. Define an arbitrary function (e.g. a Gaussian) and initialize its analytical derivative on the Chebyshev collocation points. Calculate the numerical derivative and the difference to the analytical solution. Vary the wavenumber content of the analytical function. Does it make a difference? Why is the numerical result not entirely exact?



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# IMPLEMENT YOUR SOLUTION HERE!
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Exercise 3

Now that the numerical derivative is available, we can visually inspect our results. Make a plot of both, the analytical and numerical derivatives together with the difference error.



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# PLOT YOUR SOLUTION HERE!
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