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.
86
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2
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1
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Where am I making a mistake in solving the heat equation using the spectral method (Chebyshev's differentiation matrix)?
I would like to numerically solve the following heat equation problem:
$$ u_t = \Bigg(2{a \over l}\Bigg)^2 u_{xx} \tag 1$$
$$ x \in [ -1, 1 ] \tag 2$$
$$ u(x, 0) = 0 \tag 3$$
$$ u(1, t) = A \sin \Bigg(...
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3
votes
1
answer
150
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How can we calculate mixed derivatives numerically using the Chebyshev derivative matrix?
Using the Chebyshev derivative matrix $D$, we can numerically approximate the first and second derivative of a function by doing matrix multiplication:
$${df(x) \over dx} = Df(x) \tag 1$$
$${d^2f(x) \...
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1
vote
1
answer
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in Finite Element, which approximation requires less number of unknowns: B-splines vs Shape functions vs Spectral Elements
In Finite Element Analysis, one requires to approximate the solution in terms of basis functions. My questions is, which method in general is better? by 'better', I mean which involves less number of ...
5
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1
answer
763
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Taking derivative using FFT
I would like to calculate derivative of a given function ( a 1D array) using Array. Here is the code
...
2
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0
answers
108
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Efficient heat diffusion implementation with varying coefficients
I have the following heat diffusion equation:
\begin{alignat}{3}
\partial_t u(t, \vec{x}) &= g(\vec{x})\Delta u(t,\vec{x}), &\quad& \vec{x} \in\Omega, \, t\in(0,\infty],\\
\partial_n u(t,\...
- 1,271
4
votes
2
answers
315
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Chebyshev/Lagrange polynomials in spectral methods
I am currently trying to familiarise myself with (Pseudo-)Spectral Methods for solving differential equations. Now, I am struggling to understand some obviously crucial concept of this approach. The ...
- 185
1
vote
1
answer
37
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Applying spectral method to a damped, driven 2D bending-mode wave equation on an irregular domain with heterogeneous boundary conditions
I'm trying to model some two-dimensional waves, and am unsure how to combine my boundary conditions with spectral methods. The PDE I'd like to explore resembles the equations for damped, driven ...
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3
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1
answer
158
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orthogonal basis functions on arbitrary domains and boundary conditions
I'm interested in solving an inverse coefficient problem for a PDE.
Let's say the field to be estimated is $\theta$.
The conventional approach would be to use a finite element discretization for $\...
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votes
2
answers
294
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Solving a generalized eigenvalue problem with Chebyshev spectral method: How to impose boundary conditions into the matrices?
I'm solving a local instability problem for a pipe Poiseuille flow.
The coordinate system is columnar, i.e., ($r,\theta,x$) (radial, tangential and axial).
The basic flow is $\bar{u_r}=0, \bar{u_\...
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1
vote
1
answer
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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
$$ \frac{\partial u}{\partial t} =\frac{\partial}{\partial x}(u^3\frac{\partial^3u}{\partial x^3}) $$
$$ \...
- 63
1
vote
0
answers
225
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How to discretize PDEs using Chebyshev spectral method into a system of differential algebraic equations (DAEs)?
Let's take the heat equation. We have a time derivative and spatial derivative. How to discretize the spatial derivative using Chebyshev spectral method and convert it into DAEs? Like in the form of $ ...
- 63
2
votes
1
answer
622
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Solving ODE with Spectral Method using Chebyshev Polynomials
I would like to solve following the basic equation of linear elasticity (for simplicity in 1D)
$$
\frac{d}{dx} \left( E \frac{du}{dx} \right) = 0 \quad \textrm{with} \quad u(1)=0, \; u(-1)=b
$$
...
- 23
1
vote
1
answer
285
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Solving Poisson-like PDE with FFT
Problem
I have an $n\times n$ grid, and each point on the grid is assigned two values: a score, and an (inverse) speed factor. There is a "turtle" moving along the grid, and it's goal is to ...
- 31
3
votes
1
answer
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Calculating the Jacobian for a function containing a derivative
I have the equation
$F(t) = \phi u + \frac{1}{2}\frac{d^2u}{dt^2} + u^3$
and broadly speaking, my task is to calculate the $\phi$ and $u(t)$ such that $F(t) = 0$.
I am testing out a new algorithm to ...
- 131
3
votes
2
answers
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Solving numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods
Lately, I've been trying to solve numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods.
Let $\nu$ be the viscosity and $[0,L]$ the domain. The 1D equation is,
$$
u_t + uu_x + u_{xx} ...
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1
vote
0
answers
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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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3
votes
1
answer
113
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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, ...
- 155
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votes
1
answer
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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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1
vote
2
answers
442
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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
$$...
- 219
2
votes
1
answer
200
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Pros of Fourier-Galerkin spectral methods
What are the pros of Fourier-Galerkin spectral methods while solving PDEs?
Here's the one that came in my mind first:
Easy implementation: using this method, differentiation operator computation is ...
user36293
1
vote
0
answers
194
views
Poisson equation with FFT and normalization
I'm trying to understand how to solve Poisson equation with FFT. Say, if we have the simplest periodic example
$$u_{xx}=-4\pi\cos(x)$$
The solution then should be
$$u=4\pi\cos(x)$$
I really get ...
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votes
0
answers
57
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Spectral Clustering by Andrew Ng paper: theorem proof question
I recently read the paper "On Spectral Clustering: Analysis and an Algorithm" by Ng et al.
Much of the paper centers on Theorem 2 and equation 8.
To me, it appears there is no given or referenced ...
- 11
3
votes
0
answers
69
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Spectral solver on em-pic
I'm recently studying for the spectral solver to implement EM-PIC code. I read an article and have some questions.
Many PIC codes uses spectral solver to overcome numerical artifacts on FDTD.
In the ...
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3
votes
0
answers
211
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how to construct chebyshev differentiation matrix of even/odd polynomial
I know there is efficient way to construct chebyshev differentiation matrix $D_N$. The idea is to fit the function $f(x)$ with a polynomial of $p_N(x)$ of order $N$, and use the derivative $p_N'(x)$ ...
- 31
1
vote
1
answer
535
views
Dealing with boundary conditions using Fourier spectral methods
I am currently working on a project where I need to use Fourier spectral methods to solve the KS equation. I found this code which is using the Fourier spectral methods to solve the classic 1D heat ...
user33890
1
vote
1
answer
159
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How to choose between compact finite differences and spectral methods
For a project in my advanced numerical method class I have to solve the 1D Kuramoto-Sivashinsky equation.
$$
u_t + u u_x + \lambda u_{xx} + \eta u_{xxxx} = 0.
$$
As explained here I will solve it ...
user33890
3
votes
1
answer
342
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
vote
1
answer
860
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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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votes
1
answer
816
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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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3
votes
1
answer
223
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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 ...
- 151
2
votes
1
answer
249
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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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votes
4
answers
820
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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
$$
\...
- 3,042
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1
answer
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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][1].
EDITED:...
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votes
2
answers
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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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0
answers
132
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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:
$$
(r - g(t)) v(t,z) = \pi(t,z) + (\gamma - g(t))v_z(...
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vote
1
answer
612
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Chebyshev spectral differentiation matrix for mapped domain
This is a follow up to this question.
I have mapped from Chebyshev space to the physical space using the mapping
$x = ((1-w)\xi^3 - x_p\xi^2 + w\xi + x_p + 1 )*(L/2)$.
When in Chebyshev space I have ...
- 133
1
vote
0
answers
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Find B-spline coefficients from values on collocation points
The context of my question is how to compute high order derivates on direct numerical simulation of turbulent channel flow. It is of particular interest for fluid dynamics and turbulence research.
I ...
- 121
1
vote
0
answers
62
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Pseudo-Spectral cosine transform
I'm trying to solve the following equation
$$u_t = u_{xx} + u(1-u^2), u_x(\pm 1) = 0,$$
using the Fourier cosine transform. The nonlinear term gives a convolution which I would rather avoid, which is ...
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votes
1
answer
109
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Need clarification on a piece of book excerpt about spectral element method!
I am reading "Using MPI (3rd edition)" from William Gropp, where in chapter 4 application section 4.13, it introduces an MPI application Nek5000/NekCEM which is based on spectral element method (SEM) ...
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vote
0
answers
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Perfect filtering of high frequencies in 2D FFT (Multidimensional 2/3 Rule)
Let $u_n$ be an array containing discrete values of the function $u(x,y)$. Performing a 2D FFT to this array we obtain $\hat{u_n}$ representing the values of $\hat{u}(k_x,k_y)$. I would like to ...
- 113
3
votes
0
answers
143
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Closed form PDF/CDF using Orthogonal Polynomial Expansion (gPC)
Consider a random variable which is given by an orthogonal polynomial expansion in one parameter (or polynomial chaos expansion PCE), i.e. ,
$$ f(\alpha) = \sum\limits_{n=0}^{\infty} \hat{f} (n) \psi ...
- 245
1
vote
2
answers
255
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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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votes
2
answers
193
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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 - \...
- 10.1k
3
votes
1
answer
214
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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 ...
user21030
1
vote
1
answer
736
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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3
votes
0
answers
90
views
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 $\...
- 131
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votes
0
answers
287
views
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:
...
user21030
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
...
user21030
3
votes
0
answers
122
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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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