|
[Sponsors] |
March 5, 2002, 01:09 |
Poisson eq. with Chebyshev collocation
|
#1 |
Guest
Posts: n/a
|
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 http://www.math.psu.edu/shen_j/spectral/projects/, you can download the proj2.pdf. From the second set of boundary conditions of problem 2, you may find a trick on this method !? -Paul |
|
March 6, 2002, 10:41 |
Re: Poisson eq. with Chebyshev collocation
|
#2 |
Guest
Posts: n/a
|
Hi Paul,
I had a look at your 'homework' and it seems that the second bc in probleme 2 is that u(-1)=u'(-1) 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 |
|
March 7, 2002, 02:24 |
Re: Poisson eq. with Chebyshev collocation
|
#3 |
Guest
Posts: n/a
|
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. |
|
|
|
Similar Threads | ||||
Thread | Thread Starter | Forum | Replies | Last Post |
Chebyshev collocation method | Sun Yuhui | Main CFD Forum | 3 | May 6, 2012 02:46 |
DNS Incompressible : poisson order and Div(Ui) | moomba | Main CFD Forum | 8 | May 26, 2009 08:17 |
Recommendation of a good poisson solver | Quarkz | Main CFD Forum | 2 | December 2, 2005 09:12 |
Poisson Equation in CFD | Maciej Matyka | Main CFD Forum | 9 | November 10, 2004 11:30 |
Poisson solver | Steve | Main CFD Forum | 6 | July 22, 2004 21:16 |