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Questions tagged [collocation]

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7
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1answer
141 views

Non-hermitian discretizations in quantum mechanics

Consider the Schroedinger equation $$\left(-\frac12\frac{\partial^2}{\partial x^2} + V(x) \right) \psi(x) = E \psi(x)$$ The usual way to solve it is to introduce a discretization of $\psi(x)$. This ...
1
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0answers
39 views

Numerically inverting an exponentially growing function (defined by Chebyshev polynomials)

Assume a function $M(t)$ strictly increasing, essentially growing exponentially, and asymptoptically growing at a known rate $\bar{g}$, i.e. $\lim_{t\to\infty}M'(t)/M(t) = \bar{g}$ In a set of awful ...
1
vote
1answer
231 views

Does anyone have a driver for fortran DAE solver - COLDAE?

Coldae is a solver writter by U Ascher and R Spiteri, located here http://www.cs.ubc.ca/~ascher/coldae.f . It can solve upto index-2 DAEs using collocation. I am new to Fortran, and having a hard ...
4
votes
1answer
149 views

Motivation behind Collocation Method

In previous question "Motivation behind Galerkin method", Paul gives a good and easy-understanding explanation indicating that the Galerkin method is a kind of projection method. Can anyone explain ...
1
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0answers
42 views

Second and Higher Order Order Corrector in Spectral Deferred Correction

I am trying to work out a second order or higher order correction for the method of Spectral deferred Correction (SDC). Specifically using as a corrector a second order or third order multi-step. In ...
0
votes
2answers
195 views

Calculating integrals for a function approximated by Chebyshev polynomials

Setup (complete, but all very standard): My problem is how to best calculate the cumulative integral of a function which comes out of Spectral Collocation with a chebyshev basis. Take some function $...
1
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0answers
91 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
1answer
79 views

methods for a peculiar BVP system

Consider the following system defined on the open interval (-1, 1): $y_1' = c y_3 \\ y_2' = c y_4 \\ y_3' = -f(y_1, y_2)y_2 \\ y_4' = f(y_1, y_2)y_1 $ given $ y_3(-1) = 0 = y_3(1) \\ ...
7
votes
2answers
256 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 ...
4
votes
0answers
84 views

How do I perform chebyshev interpolation from a to b with custom angle range?

Typically Chebyshev interpolation from $-1$ to $1$ with angle from $0$ to $\pi$: $\xi_j=\cos \left ({\pi j \over N}\right )$ $x_j=(1+\xi_j) * {L \over 2}$ $w$: $w_0=\pi/(2N)$ $w_{1,...,N-1}=\pi/(N)$...
2
votes
1answer
93 views

Sparse matrices origins

I am using the sparse matrices provided by the University of Florida Sparse Matrix Collection and most matrices are accompanied with little description of the problem or discipline from which the ...
3
votes
0answers
81 views

Stochastic Collocation for time evolving ODE

For an Stochastic Differential Equation, e.g., $$ \frac{du}{dt} = \alpha*\sin(u*t) $$ where $\alpha$ is normally distributed with nonzero mean, I am trying to use a stochastic collocation approach ...
4
votes
2answers
597 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 ...
2
votes
2answers
312 views

Numerical solution of fractional integro-diffrential equ. using collocation method?

problem comes from "Numerical solution of fractional integro-differential , equations by collocation method , E.A. Rawashdeh, Department of Mathematics, Yarmouk University, Irbid 21110, Jordan" $D^...
3
votes
1answer
723 views

Where can I find coded examples of stochastic collocation applied to an elliptical PDE using smolyak sampling?

I'm having some troubles implementing a collocation method to solve a stochastic partial differential equation of the form: $\nabla (a(x,w)\nabla u(x,w))=f(x,w)$ in $D$, $u=g$ in $\partial D$ where $...