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Paul March 5, 2002 02:09

Poisson eq. with Chebyshev collocation
Dear all, I am trying to solve Poisson equation with Chebyshev collocation method. It is essentially a direct method, via two multiplications on the eigenvetors. However, I can only solve it with following b.c (in either direction)

1) Both Dirichlet 2) One Dirichlet and one Robin\

For Neumann conditions, I always fails. If anyone has experience in this topic, please kindly provide your comments.

P.S. from this link, you can download the proj2.pdf. From the second set of boundary conditions of problem 2, you may find a trick on this method !?


Patrick Godon March 6, 2002 11:41

Re: Poisson eq. with Chebyshev collocation
Hi Paul,

I had a look at your 'homework' and it seems that the second bc in probleme 2 is that


or that the function is equal to its own derivative... I am not sure what physical sens this makes and it does not seem very clear, unless u(-1)=u'(-1)=0 ....

However, it is a condition on u(-1) not on u'(-1). And the main problem here would be that you have to compute the derivative of the function (u') at the boundary point and then impose this value on u at that boundary, and this would not be self-consistent.. an easy way would just be to change a0 in the expansion of u, such that u=u'.

However, the best way to do it, is as follows: Usually if you have a bc like u'=0, you change the value of u at the boundary such that u'=0. This is done by writting u'(xi) as a function of u(xj). Then you get one equation u'(xi=-1) = 0 (xi=-1, i=0), this gives you a relation between all the u(xj), j=0, 1, ..N. You then solve for u(xj=-1) as a function of the other u(xj). Here, I guess you could also solve for u'=u, u'(xi=-1)=u(xj=-1), you will have one equation where u(xj=1) appears on both sides, and you will have to solve this explicitly for u(xj=-1).

You have to bear in mind that the value that you impose in this particular case on u(x=-1) is computed from the numerical probleme, it is not a fixed value like 0, -1, -5, or whatever. Thus if there are small numerical oscillations, and you solve also as a function of time, these oscillations can amplify. Fortunately it is not the case here.

Let me know if I was clear enough,

Cheers, Patrick

Paul March 7, 2002 03:24

Re: Poisson eq. with Chebyshev collocation
Thanks Patrick. The algorithm your described is very effecient in treating Helmholtz equtions occuring in solving NSe. For that equation, I always can get correct solution (I see). What I am really concerned is the Poisson equation which with the Neumann boundary conditon may pose a ill-condition matrix. But unfortunately, I can not find a reference EXACT on this topic. If you can make time, try to solve this simple 1D Poisson equation with Direct Chebyshev collocation method

u_xx = f in x=[-1,1] and f = - pi^2 sin(pi*(x-0.5))

exact solution: u = sin(pi*(x-0.5)); at x=-1, u_x = 0; at x=1, u=1.

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