# Questions tagged [spectral-method]

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### 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 ...
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 ...
692 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 ...
323 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 ...
927 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 ...
222 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 ...
654 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, ...
2k views

220 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 $f(t_i, \cdot)$ ...
855 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 $u_t$. ...
884 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. ...
633 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 ? ...
108 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 ...
169 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 ...
151 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: $\nabla\cdot 2\mu(\dot\varepsilon)\dot\varepsilon - \... 1answer 75 views ### Solving new linear system that comes from an$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 $$... 1answer 454 views ### 2d pseudo-spectral turbulence simulation with random initial velocities I am trying to write a 2d pseudo-spectral DNS code with random initial velocities. This is kind of a classic simulation where the very tiny vortices group together forming larger and larger vortices ... 1answer 125 views ### Trignometric/Fourier spectral collocation with zero Dirichlet BC in 2D I am concerned with numerical solution to the following problem on [0,1]\times[0,1]. \dfrac{\partial\theta}{\partial t}+u(x,t).\nabla \theta(x,t)=\kappa \nabla^2\theta(t,x) with Dirichlet ... 1answer 916 views ### Partial derivatives of a 3D array in Matlab I'm interested in taking some partial derivatives of a 3 dimensional array in Matlab - say A(i,j,k) approximates f(x_i,y_j,z_k). I need to approximate things like \partial_{xy}f, \partial_{yz}... 1answer 152 views ### Average value divergence in spectral method for Poisson equation I'd like to know how to deal with a divergence when trying to solve the Poisson equation for electrostatics with a simple spectral method. I'm not sure how to best state my problem, so I'll explain ... 1answer 280 views ### How to compute all the eigenvalues of a large sparse matrix using matlab? In matlab, there are 2 commands named "eig" for full matrices and "eigs" for sparse matrices to compute eigenvalues of a matrix. And eig(A) computes all the eigenvalues of a full matrix and eigs(A) ... 2answers 248 views ### Solve wave equation with discontinuous coefficients numerically? I would like to solve the following equation$$\frac{\partial^2 y}{\partial t^2} - c^2(x,t)\frac{\partial^2 y}{\partial x^2}=0,$$for y=y(x,t) numerically. The wave speed, c(x,t), is of the form$$... 3answers 716 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) - \...
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 ...