Questions tagged [spectral-method]

For questions about spectral methods, a technique for solving differential equations by expressing them in terms of some computationally convenient basis (typically that obtained via fast fourier transform). Questions could relate to the theory behind the method or details of implementing for a particular problem.

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Solving a generalized eigenvalue problem with Chebyshev spectral method: How to impose boundary conditions into the matrices?

I'm solving a local instability problem for a pipe Poiseuille flow. The coordinate system is columnar, i.e., ($r,\theta,x$) (radial, tangential and axial). The basic flow is $\bar{u_r}=0, \bar{u_\...
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What does the Chebyshev differentiation matrix look like for third and fourth derivative?

I have a PDE that contains both the 3rd derivative and 4th derivative. Example shown below $$ \frac{\partial u}{\partial t} =\frac{\partial}{\partial x}(u^3\frac{\partial^3u}{\partial x^3}) $$ $$ \...
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1 vote
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How to discretize PDEs using Chebyshev spectral method into a system of differential algebraic equations (DAEs)?

Let's take the heat equation. We have a time derivative and spatial derivative. How to discretize the spatial derivative using Chebyshev spectral method and convert it into DAEs? Like in the form of $ ...
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1 vote
1 answer
147 views

Solving ODE with Spectral Method using Chebyshev Polynomials

I would like to solve following the basic equation of linear elasticity (for simplicity in 1D) $$ \frac{d}{dx} \left( E \frac{du}{dx} \right) = 0 \quad \textrm{with} \quad u(1)=0, \; u(-1)=b $$ ...
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1 answer
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Solving Poisson-like PDE with FFT

Problem I have an $n\times n$ grid, and each point on the grid is assigned two values: a score, and an (inverse) speed factor. There is a "turtle" moving along the grid, and it's goal is to ...
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3 votes
1 answer
98 views

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 ...
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2 answers
639 views

Solving numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods

Lately, I've been trying to solve numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods. Let $\nu$ be the viscosity and $[0,L]$ the domain. The 1D equation is, $$ u_t + uu_x + u_{xx} ...
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Finite element method for an equation requiring switch between spectral and temporal domain

Some equations (such as the non-linear schrödinger equation for pulse propagation) are more easily solved in the spectral form, but still need a representation in the temporal domain to calculate ...
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1 answer
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How does the diffusion of a finite volume method with a WENO scheme compare with that of spectral methods?

I know that, in general, finite volume (FV) methods are more (numerically) diffusive than spectral methods. However, I can't find any information on how the advection scheme changes that. For example, ...
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1 answer
768 views

Gauss-Lobatto quadrature and nodal points for FEM

By using the Legendre-Gauss-Lobatto (LGL) quadrature formula (QF) and LGL nodal points one achives a diagonal mass-matrix for finite element problems. (More specifically, the spectral element method.) ...
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Solve wave equation with discontinuous coefficients numerically?

I would like to solve the following equation $$\frac{\partial^2 y}{\partial t^2} - c^2(x,t)\frac{\partial^2 y}{\partial x^2}=0,$$ for $y=y(x,t)$ numerically. The wave speed, $c(x,t)$, is of the form $$...
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1 answer
132 views

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 ...
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1 vote
0 answers
130 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 ...
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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 ...
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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 ...
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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 $p_N'(x)$ ...
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1 answer
211 views

Dealing with boundary conditions using Fourier spectral methods

I am currently working on a project where I need to use Fourier spectral methods to solve the KS equation. I found this code which is using the Fourier spectral methods to solve the classic 1D heat ...
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1 vote
1 answer
111 views

How to choose between compact finite differences and spectral methods

For a project in my advanced numerical method class I have to solve the 1D Kuramoto-Sivashinsky equation. $$ u_t + u u_x + \lambda u_{xx} + \eta u_{xxxx} = 0. $$ As explained here I will solve it ...
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4 votes
1 answer
273 views

What is the difference between Methods of Weighted Residuals and Spectral Methods?

Methods of Weighted Residuals (MWR) [1] usually include Galerkin, collocation, method of moments, least-squares and subdomain methods. Spectral methods [2] usually include Galerkin, tau and ...
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1 answer
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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) ...
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Need an example Legendre-Gauss-Radau pseudospectral differentiation matrix or Matlab code

I'm trying to implement various kinds of pseudospectral methods for direct optimization in Matlab using IPOPT. I've got some working Legendre-Gauss-Lobatto code, but would like to use the flipped ...
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1 answer
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Imposition of Dirichlet BC for Fourier pseudospectral in this paper

I was trying to implement the algorithm from the paper "Adapting a Fourier pseudospectral method to Dirichlet boundary conditions for Rayleigh–Benard convection". I am having a hard time to ...
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2 votes
1 answer
187 views

Average value divergence in spectral method for Poisson equation

I'd like to know how to deal with a divergence when trying to solve the Poisson equation for electrostatics with a simple spectral method. I'm not sure how to best state my problem, so I'll explain ...
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2 votes
1 answer
218 views

build a simple incompressible solver with spectral method

I am trying to build a simple solver for the incompressible fluid in a periodic box to learn the spectral method. I am following the textbook (Peyret R. Spectral methods for incompressible viscous ...
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4 votes
4 answers
630 views

Role of weight function in Galerkin methods

I have difficulties in understanding the role of the weight function $w(x)$ that occurs in the solution of PDEs via the Galerkin approach. Consider a linear differential equation of the form $$ \...
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1 vote
2 answers
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Continuum removal algorithm in python

I'm using python 2.7 (on jupyter notebook, win10 64 bit) to perform my analysis. I need to perform continuum removal (CR) on a reflectance spectrum data. I need it to be as described here. EDITED: ...
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13 votes
2 answers
1k views

Why can ill-conditioned linear systems be solved precisely?

According to the answer here, large condition number (for linear system solving) decreases the guaranteed number of correct digits in the floating point solution. Higher order differentiation matrices ...
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1 vote
0 answers
126 views

Solve a PDE BVP with Spectral methods in time and space?

I have a PDE (coming from a Bellman equation with $z$ under brownian motion). Let $z \in [0,\infty)$ and $t \in [0,T]$. To sketch the equation: $$ (r - g(t)) v(t,z) = \pi(t,z) + (\gamma - g(t))v_z(...
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1 vote
1 answer
501 views

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 ...
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1 vote
0 answers
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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 ...
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0 answers
57 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 ...
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1 answer
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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) ...
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1 vote
0 answers
58 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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3 votes
0 answers
140 views

Closed form PDF/CDF using Orthogonal Polynomial Expansion (gPC)

Consider a random variable which is given by an orthogonal polynomial expansion in one parameter (or polynomial chaos expansion PCE), i.e. , $$ f(\alpha) = \sum\limits_{n=0}^{\infty} \hat{f} (n) \psi ...
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1 vote
2 answers
216 views

How is the dense system usually dealt with in spectral method?

Unlike finite element (FEM) or finite difference methods (FDM), where the original PDE is transformed into a sparse linear system, spectral methods return a dense linear system. For a large system, it'...
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2 votes
2 answers
173 views

Finite element method in $x$, $y$, spectral method in $z$

I'm working on a certain problem of slow, non-Newtonian, thin-film flow. This problem can be modelled with the incompressible Stokes equations: $\nabla\cdot 2\mu(\dot\varepsilon)\dot\varepsilon - \...
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3 votes
1 answer
202 views

How to define a non-square Legendre pseudospectral differentiation matrix?

I am going to discuss my reasons for wanting this first, as this may in fact not be what I am looking for. My reason for asking this that I have finished writing a piece of code that solves, $-\nabla ...
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2 votes
1 answer
661 views

Spectral methods, Spectral Volume methods, Spectral Difference methods

Could someone explain the link (if any) between the spectral methods (SM), as presented for example here and the so called spectral volume methods (SV) and spectral difference methods (SD) for CFD ? ...
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0 answers
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Differential eigenproblem with eigenvalue in boundary condition

Statement of the problem I need to (numerically) solve an eigenproblem of the type $$-\omega^2\mathcal{D}_1\vec{x}=\mathcal{D}_2\vec{x}$$ on the interval $[-1,1]$, where $\mathcal{D}_1$ and $\...
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How to form the stiffness matrix for the Poisson equation using a spectral method

This is a follow-up question to How do I form the Chebyshev differentiation matrix in MATLAB? The goal of the following code is to solve the Poisson problem: ...
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3 votes
1 answer
2k views

How do I form the Chebyshev differentiation matrix in MATLAB?

I have some code that does exactly this, but I do not like to use things I do not understand. Here is the code ...
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3 votes
0 answers
110 views

Spectral/hp-finite elements for 4th order PDEs

Does anyone know of references discussing the solution of 4th order PDEs by way of spectral/hp-finite element methods? Specifically, I'm interested in the extension of the spectral element method, ...
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5 votes
2 answers
294 views

Motivation behind Collocation Method

In the previous question "Motivation behind Galerkin method", Paul gives a good and easy-to-understand explanation indicating that the Galerkin method is a kind of projection method. Can anyone ...
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3 votes
1 answer
190 views

Achieving high relative accuracy (vs. absolute accuracy) using spectral methods

Problem I have a PDE that I'm trying to solve with spectral methods. The solution $y$ is always positive, and decays as $y \propto e^{-ax}$ for large $x$. The domain is $[0, \infty)$. (There are ...
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1 vote
0 answers
59 views

Second and Higher Order Order Corrector in Spectral Deferred Correction

I am trying to work out a second order or higher order correction for the method of Spectral deferred Correction (SDC). Specifically using as a corrector a second order or third order multi-step. In ...
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2 votes
1 answer
940 views

Numerically computing the advection equation

I am trying to write a program to compute the advection equation. $$u_t +u_x = 0$$ I use the spectral method for the spatial derivative $u_x$ and the leapfrog method for the time derivative $u_t$. ...
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4 votes
2 answers
472 views

Interpolation with the roots of orthogonal polynomials & Spectral expansion

I'm a bit confused about the relationships between these two approximation methods mentioned in the title. Does this kind of interpolation also belongs to the field of spectral methods? Are the ...
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2 votes
1 answer
979 views

How can i get gauss-lobatto points on a quadrilateral?

How can I get Gauss-Lobatto points on a quadrilateral or a triangle in $x$-$y$ plane? I am only getting abscissa coordinates and weights by solving Lobatto polynomials using Lobatto quadrature. ...
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2 votes
1 answer
75 views

Solving new linear system that comes from an $p$ enrichment

Let's say I am solving a simple Poisson problem using a Mixed (DG) finite element method. If we use orthogonal polynomials as basis functions we can write the finite-dimensional linear system as $$ ...
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3 votes
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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 ...
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