Questions tagged [spectral-method]

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19
votes
1answer
625 views

Difficulty with Spectral Method using Chebyshev Polynomials

I am having a bit of difficulty in trying to understand a paper. The paper uses spectral method to solve for an eigenvalue that comes from a system of coupled ODEs. I will write out only one equation ...
13
votes
2answers
640 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 ...
9
votes
2answers
612 views

Boundary conditions Chebyshev differentiation

I was wondering if anyone has any experience dealing with boundaries when implementing chebyshev differentiation. I am currently trying to implement a no slip boundary condition to solve the ...
7
votes
2answers
283 views

Spectral Methods in time

I was reading up on Spectral Methods for PDEs. In all the descriptions I read, while the position component is approximated via a Fourier series or other methods, the time component is still ...
7
votes
3answers
852 views

Conforming mesh refinement for quads/hex elements

The context - I'm working with a spectral FE (higher order interpolation at GLL nodes) code on conforming hexahedral meshes, and our PI is interested in improving mesh quality, possibly with adaptive ...
7
votes
2answers
208 views

Enforcing non-negative constraint in fourier-spectral method

I have a PDE optimization problem, and a scalar field (which I am optimizing over) is supposed to be nonnegative everywhere in the domain. Since I am working in Fourier space for solving this problem ...
6
votes
0answers
617 views

Stochastic Galerkin projection approach for using generalized polynomial chaos expansion (GPCE) in solving PDE

I want to know if there is any way to define the test and trial function in the way that I want instead of using the default functions. So if I want define the polynomial and basis and coefficient, ...
5
votes
1answer
2k views

Sign differences in spectral decomposition in NumPy

I am trying to understand an example from a book, but I seem to get different answers depending on which spectral decomposition function I use in NumPy. I am trying to find a spectral decomposition $...
5
votes
3answers
420 views

Variable viscosity Stokes equation

One very efficient way to solve Stokes equation with periodic boundary conditions \begin{equation} -\eta \nabla^{2} \bf{v} + \nabla p = f \\ \nabla \cdot \bf{v} = 0 \end{equation} is using the ...
4
votes
4answers
296 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 $$ \...
4
votes
2answers
242 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 ...
4
votes
2answers
431 views

How are Spectral Methods applied to CFD? In particular, how is the pressure-velocity coupling implemented?

I know that spectral-methods which are weighted-residual methods can be applied to solve for the incompressible N-S equations. In particular, they are applied to Direct Numerical Simulations (DNS) for ...
4
votes
2answers
633 views

Orthonormalized Bernstein polynomials using Gram-Schmidt

I was wondering, before trying to do that myself, has anyone attempted to do orthonormalization of Bernstein polynomials using Gram-Schmidt? I discussed this with several people and have been told ...
4
votes
1answer
225 views

How to calculate numerical dispersion relations for Spectral Elements?

How can we determine the numerical dispersion relation of a Spectral Element Method which leads to coupled systems of algebraic equations? What approaches to do analysis of dispersion relations is ...
4
votes
1answer
819 views

Chebyshev spectral differentiation via FFT

I am using the Chebyshev spectral differentiation technique that is described concisely under "details" here. The idea is to take the initial data $v_0,v_1\,...,v_N$ and store it in union with itself ...
4
votes
1answer
98 views

Zero-k mode in Pseudo-spectral solution of Stokes Flow

I'm trying to solve a Stokes flow problem with a pseudo-spectral method in periodic boundary conditions. The equations of interest are $-\nabla^2 \bf{v} + \nabla p = \bf{f} \\ \nabla \cdot \bf{v} = ...
4
votes
1answer
130 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 ...
4
votes
1answer
503 views

Two approaches to solving diffusion equation in Fourier space

I want to numerically solve the diffusion equation $\partial_t u = D \partial_x^2 u$ in Fourier space, and can think of multiple ways to do it. Setup Option 1 Differentiating $u$ twice in Fourier ...
4
votes
0answers
65 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 $\...
3
votes
2answers
302 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 ...
3
votes
3answers
819 views

Solving Stokes flow with walls using Oseen tensor

Introduction I've developed a code to solve for generalised, incompressible 2D Stokes flow $\eta \nabla^2 \mathbf{v} - \nabla p + \mathbf{S} = 0$ $\nabla . \mathbf{v} = 0$ where $\mathbf{S}$ can ...
3
votes
1answer
1k 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 ...
3
votes
1answer
195 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 ...
3
votes
1answer
171 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 ...
3
votes
0answers
107 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 ...
3
votes
0answers
101 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, ...
3
votes
0answers
196 views

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 ...
3
votes
0answers
179 views

Integration of nonlinear PIDE via spectral methods

At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction $\...
3
votes
0answers
198 views

Choosing good basis functions to approximate a Lipschitz function

Let $D = \left\{0, t_1, t_2, \ldots, t_n\right\} \times [0,1]$ and $$ f: D\to [0,1], $$ be a function of time and a one-dimensional space. There is no analytical formula for $f$, but $f(t_i, \cdot)$ ...
2
votes
1answer
711 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 $u_t$. ...
2
votes
1answer
747 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. ...
2
votes
1answer
581 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 ? ...
2
votes
1answer
120 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 ...
2
votes
2answers
125 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 - \...
2
votes
1answer
73 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 $$ ...
2
votes
1answer
418 views

2d pseudo-spectral turbulence simulation with random initial velocities

I am trying to write a 2d pseudo-spectral DNS code with random initial velocities. This is kind of a classic simulation where the very tiny vortices group together forming larger and larger vortices ...
2
votes
1answer
121 views

Trignometric/Fourier spectral collocation with zero Dirichlet BC in 2D

I am concerned with numerical solution to the following problem on $[0,1]\times[0,1]$. $\dfrac{\partial\theta}{\partial t}+u(x,t).\nabla \theta(x,t)=\kappa \nabla^2\theta(t,x)$ with Dirichlet ...
2
votes
1answer
813 views

Partial derivatives of a 3D array in Matlab

I'm interested in taking some partial derivatives of a 3 dimensional array in Matlab - say $A(i,j,k)$ approximates $f(x_i,y_j,z_k)$. I need to approximate things like $\partial_{xy}f$, $\partial_{yz}...
1
vote
1answer
86 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) ...
1
vote
3answers
582 views

Solve diffusion equation with linear source term

I would like to solve numerically the diffusion equation, where the sink term depends linearly on the field, and there is field-independent sink: $\frac{\partial^2 u(x)}{\partial x^2} =f(x)u(x) - \...
1
vote
1answer
69 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 ...
1
vote
1answer
312 views

Solve steady state reaction-diffusion/Helmholtz equation numerically

I am solving a problem of the form: $\dfrac{\partial u(x,y,t)}{\partial t} = \nabla^2 u(x,y,t) - f(x,y,t)u(x,y,t) - \kappa(x,y,t)$ At the moment, I am solving this at each time step by assuming a ...
1
vote
2answers
1k views

Computing Kolmogorov/Energy spectrum for turbulent boundary layer

Previously, I have calculated energy spectrum for 3D DNS data obtained for isotropic turbulence which is equally spaced in all three directions and then to compute the energy spectrum, one performs ...
1
vote
1answer
46 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 ...
1
vote
2answers
133 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'...
1
vote
1answer
372 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 ...
1
vote
1answer
88 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 ...
1
vote
0answers
77 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(...
1
vote
0answers
75 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 ...
1
vote
0answers
39 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 ...