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7
votes
2answers
153 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 ...
0
votes
0answers
34 views

Implicit time integrator for Chebyshev collocation method for linear hyperbolic system

I want to solve linear hyperbolic system using Chebyshev collocation method. As this method puts severe constraint on the time step for the explicit time integration, I decided to switch to implicit ...
1
vote
0answers
24 views

Aliasing question for a spectrally derived Lagrangian

Consider two functions $$X=\sum_{n=-N}^N i\ sign(n) y_n e^{-in\xi}; \quad \quad Y = \sum_{n=-N}^N y_n e^{-in\xi}.$$ where $y_n$ are the dependent variables of the system, and are functions of time. ...
1
vote
0answers
53 views

Am I using the incorrect implementation of the fast Chebyshev transform?

I was told that the fast Chebyshev transform has superior spectral convergence, but I am unable to verify its rumored convergence. I was given plots of its spectral convergence, where the signal's ...
4
votes
1answer
94 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
3answers
145 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 ...
3
votes
1answer
64 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} = ...
1
vote
1answer
143 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
3answers
160 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) - ...
-2
votes
1answer
208 views

explain the difference between 1D Poisson solvers

I compared 2 methods for 1 dimensional Poisson equation solution. One is finite-difference method, "Successive Overrelaxation" from http://www.cs.berkeley.edu/~demmel/cs267/lecture24/lecture24.html; ...
2
votes
1answer
75 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 ...
3
votes
1answer
193 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 ...
4
votes
1answer
85 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 ...
2
votes
0answers
130 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 ...
2
votes
3answers
268 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 ...
1
vote
2answers
418 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 ...
3
votes
1answer
322 views

spectral decomposition in Numpy, sign difference

I am trying to follow along with an example from a book, but I get seemingly different answers depending on which spectral decomposition function I use in Numpy. I am trying to transform the Matrix G, ...
2
votes
1answer
182 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 ...
7
votes
2answers
190 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 ...
2
votes
1answer
411 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 $\partial_{xy}f$, ...
3
votes
0answers
121 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 $f(t_i, \cdot)$ ...
6
votes
3answers
371 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 ...
0
votes
1answer
35 views

a question about kernelized locality preserving projections

kernel LPP is of form: $$\min_{\alpha} \ \alpha^{T}KLK\alpha \\ s.t. \ \alpha^{T}KDK\alpha = 1$$ and it eventually results in solving generalized eigenvalue problem below: $$KLK \alpha= \lambda KDK ...
5
votes
0answers
321 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, ...
4
votes
1answer
396 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
2answers
343 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 ...
19
votes
1answer
515 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 ...