# 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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### 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 ...
763 views

### Taking derivative using FFT

I would like to calculate derivative of a given function ( a 1D array) using Array. Here is the code ...
108 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,\...
315 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 ...
1 vote
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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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1 vote
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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 ...
113 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, ...
1k 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.) ...
1 vote
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1 vote
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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:...
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 ...
1 vote
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1 vote
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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'...
193 views

I'm working on a certain problem of slow, non-Newtonian, thin-film flow. This problem can be modelled with the incompressible Stokes equations: $\nabla\cdot 2\mu(\dot\varepsilon)\dot\varepsilon - \... 3 votes 1 answer 214 views ### How to define a non-square Legendre pseudospectral differentiation matrix? I am going to discuss my reasons for wanting this first, as this may in fact not be what I am looking for. My reason for asking this that I have finished writing a piece of code that solves,$-\nabla ... 1 vote
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### 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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### 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 \$\...
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: ...  