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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What is the advantage of using a particular RK Scheme?

The Wikipedia article on Runge-Kutta Methods lists several examples of each order. My question is, are there any particular advantages using one particular scheme over another of the same order? I ...
Jacob Ivanov's user avatar
0 votes
0 answers
47 views

How to correctly implement boundary conditions with Chebyshev differentiation matrix?

In my code I am able to implement zero Dirichlet boundary conditions, however, in my opinion my method does not seem to be as smooth and effective as possible. I am solving a Poisson's equation with ...
Jamie 's user avatar
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1 vote
0 answers
68 views

Where is the issue in my nonlinear PDE solver using pseudo-spectral methods with Chebyshev collocation?

This is a repost from another community, I think the question would be better here. This bug has been haunting me for a while and would love new pair of eyes to help, not sure if this is the right ...
Jamie 's user avatar
  • 121
2 votes
0 answers
78 views

Preventing an Overflow in Exponential Integrating Factor

The following is the Discrete Spectral Vorticity Evolution PDE for incompressible flow: $$ \frac{\partial \Omega_{pq}}{\partial t} = \nu \left( \frac{\partial^2 \Omega_{pq}}{\partial x^2} + \frac{\...
Jacob Ivanov's user avatar
2 votes
1 answer
218 views

What is the correct way to implement Neumann type boundary conditions for solving a PDE using Chebyshev's collocation method?

I am studying spectral methods for solving PDEs numerically. I finished a chapter that explains how to use Chebyshev's collocation method to solve them. Though the explanation in the book is quite ...
Amilox Lex's user avatar
2 votes
1 answer
283 views

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(...
FriendlyNeighborhoodEngineer's user avatar
3 votes
1 answer
167 views

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) \...
FriendlyNeighborhoodEngineer's user avatar
1 vote
1 answer
108 views

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 ...
Hosein Javanmardi's user avatar
5 votes
1 answer
808 views

Taking derivative using FFT

I would like to calculate derivative of a given function ( a 1D array) using Array. Here is the code ...
learning_physics's user avatar
2 votes
0 answers
110 views

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,\...
lightxbulb's user avatar
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4 votes
2 answers
385 views

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 ...
Octavius's user avatar
  • 185
1 vote
1 answer
48 views

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 ...
MRule's user avatar
  • 153
3 votes
1 answer
172 views

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 $\...
Daniel Shapero's user avatar
0 votes
2 answers
417 views

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_\...
Jack's user avatar
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1 vote
1 answer
380 views

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 $$ \frac{\partial u}{\partial t} =\frac{\partial}{\partial x}(u^3\frac{\partial^3u}{\partial x^3}) $$ $$ \...
dazemood's user avatar
1 vote
0 answers
237 views

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 $ ...
dazemood's user avatar
2 votes
1 answer
702 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) $$ \frac{d}{dx} \left( E \frac{du}{dx} \right) = 0 \quad \textrm{with} \quad u(1)=0, \; u(-1)=b $$ ...
PS-Elas's user avatar
  • 23
1 vote
1 answer
313 views

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 ...
programjames's user avatar
3 votes
1 answer
120 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 ...
Paddy's user avatar
  • 131
3 votes
2 answers
2k 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} ...
Mathieu Rousseau's user avatar
1 vote
0 answers
60 views

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 ...
arc_lupus's user avatar
  • 553
3 votes
1 answer
125 views

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, ...
TomCho's user avatar
  • 155
6 votes
1 answer
2k 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.) ...
Dagon's user avatar
  • 128
1 vote
2 answers
467 views

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 $$...
Peanutlex's user avatar
  • 219
2 votes
1 answer
215 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 ...
user avatar
1 vote
0 answers
208 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 ...
Tangur's user avatar
  • 11
0 votes
0 answers
58 views

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 ...
Koby Hayashi's user avatar
3 votes
0 answers
69 views

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 ...
asdgaaa1123's user avatar
3 votes
0 answers
221 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 $p_N'(x)$ ...
user34337's user avatar
1 vote
1 answer
644 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 ...
user avatar
1 vote
1 answer
165 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 ...
user avatar
3 votes
1 answer
349 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 ...
L. Young's user avatar
  • 188
1 vote
1 answer
970 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) ...
Happy's user avatar
  • 961
0 votes
1 answer
844 views

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 ...
lamont's user avatar
  • 143
0 votes
1 answer
176 views

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 ...
user162281's user avatar
3 votes
1 answer
223 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 ...
lnmaurer's user avatar
  • 151
2 votes
1 answer
268 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 ...
user28005's user avatar
4 votes
4 answers
868 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 $$ \...
davidhigh's user avatar
  • 3,127
1 vote
1 answer
3k views

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:...
user88484's user avatar
  • 121
13 votes
2 answers
2k 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 ...
Zoltan Csati's user avatar
1 vote
0 answers
136 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(...
jlperla's user avatar
  • 376
1 vote
1 answer
626 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 ...
Rhinocerotidae's user avatar
1 vote
0 answers
93 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 ...
Hydro Guy's user avatar
  • 121
1 vote
0 answers
67 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 ...
Gregory's user avatar
  • 184
0 votes
1 answer
109 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) ...
user123's user avatar
  • 679
1 vote
0 answers
77 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 ...
Bremsstrahlung's user avatar
3 votes
0 answers
143 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 ...
Amir Sagiv's user avatar
1 vote
2 answers
265 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'...
user123's user avatar
  • 679
2 votes
2 answers
196 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 - \...
Daniel Shapero's user avatar
3 votes
1 answer
215 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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