# Questions tagged [spectral-method]

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### Calculating the Jacobian for a function containing a derivative

I have the equation $F(t) = \phi u + \frac{1}{2}\frac{d^2u}{dt^2} + u^3$ and broadly speaking, my task is to calculate the $\phi$ and $u(t)$ such that $F(t) = 0$. I am testing out a new algorithm to ...
1answer
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### Pros of Fourier-Galerkin spectral methods

What are the pros of Fourier-Galerkin spectral methods while solving PDEs? Here's the one that came in my mind first: Easy implementation: using this method, differentiation operator computation is ...
0answers
102 views

### Poisson equation with FFT and normalization

I'm trying to understand how to solve Poisson equation with FFT. Say, if we have the simplest periodic example $$u_{xx}=-4\pi\cos(x)$$ The solution then should be $$u=4\pi\cos(x)$$ I really get ...
0answers
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### Spectral Clustering by Andrew Ng paper: theorem proof question

I recently read the paper "On Spectral Clustering: Analysis and an Algorithm" by Ng et al. Much of the paper centers on Theorem 2 and equation 8. To me, it appears there is no given or referenced ...
1answer
280 views

### How to compute all the eigenvalues of a large sparse matrix using matlab?

In matlab, there are 2 commands named "eig" for full matrices and "eigs" for sparse matrices to compute eigenvalues of a matrix. And eig(A) computes all the eigenvalues of a full matrix and eigs(A) ...
0answers
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### Spectral solver on em-pic

I'm recently studying for the spectral solver to implement EM-PIC code. I read an article and have some questions. Many PIC codes uses spectral solver to overcome numerical artifacts on FDTD. In the ...
0answers
120 views

### how to construct chebyshev differentiation matrix of even/odd polynomial

I know there is efficient way to construct chebyshev differentiation matrix $D_N$. The idea is to fit the function $f(x)$ with a polynomial of $p_N(x)$ of order $N$, and use the derivative $p_N'(x)$ ...
4answers
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1answer
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### Chebyshev spectral differentiation matrix for mapped domain

This is a follow up to this question. I have mapped from Chebyshev space to the physical space using the mapping $x = ((1-w)\xi^3 - x_p\xi^2 + w\xi + x_p + 1 )*(L/2)$. When in Chebyshev space I have ...
0answers
82 views

### Find B-spline coefficients from values on collocation points

The context of my question is how to compute high order derivates on direct numerical simulation of turbulent channel flow. It is of particular interest for fluid dynamics and turbulence research. I ...
0answers
51 views

### Pseudo-Spectral cosine transform

I'm trying to solve the following equation $$u_t = u_{xx} + u(1-u^2), u_x(\pm 1) = 0,$$ using the Fourier cosine transform. The nonlinear term gives a convolution which I would rather avoid, which is ...
1answer
101 views

### Need clarification on a piece of book excerpt about spectral element method!

I am reading "Using MPI (3rd edition)" from William Gropp, where in chapter 4 application section 4.13, it introduces an MPI application Nek5000/NekCEM which is based on spectral element method (SEM) ...
0answers
52 views

### Perfect filtering of high frequencies in 2D FFT (Multidimensional 2/3 Rule)

Let $u_n$ be an array containing discrete values of the function $u(x,y)$. Performing a 2D FFT to this array we obtain $\hat{u_n}$ representing the values of $\hat{u}(k_x,k_y)$. I would like to ...
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### Spectral Collocation (or Weighted Residual) Methods to solve Stiff ODEs?

I have a system of ODEs which is (at least moderately) stiff. Consider the class of spectral collocation methods https://en.wikipedia.org/wiki/Spectral_method or the related class of weighted ...