Numerical Solutions for Mixed Boundary Condition

tau1777

Hi All,

I was reading this paper the other day and I've been trying to find the numerical techniques its mentions but have been thus far unsuccessful. The authors simply state that is well know and straightforward, and they believe this so much that they don't even include a reference. Ok, sorry about the rant.

The general problem they are trying to solve is for the r-mode oscillations of Neutron Stars. They get everything down to a 2nd order 1-d differential equation. They say the solution is zero at r=0, and at the surface they say it obeys something like A[r] * δρ[r] + B[r] * ∂ δρ[r] / ∂r =0. They they say they integrate from r=0 with the condition that δρ [0] =0, and from the surface with the condition A[r] * δρ[r] + B[r] * ∂ δρ[r] / ∂r =0 and they match the solutions at some specified point, and they use the frequency of the mode as the parameter they mess with to match the solution.

I understand how one can integrate out to the surface with the condition that δρ [0] , but how do they does the integration from the surface to the interior work when one has a Mixed boundary condition?

Any help is greatly appreciated. Thanks.

hunt_mat

One possible method is Newton's method but with this methods you have to made an initial stab at the solution in the first place (asymptotic analysis perhaps?) It works very well when it works and then you can just code up the BCs without a problem.

I have some initial matlab code you can have if you want.

Mat

tau1777

Hey Mat,

Thanks very much for the response. So if I understand what you are saying. I should just finite difference the boundary condition have have a an algebraic equation for the boundary. Then solve this equation and all the equations I got from the interior using a Newton-Raphson iteration scheme. And I need to find some way to find an initial guess for the iteration. Is this what you mean?

I'm basically trying to do something like this now within Mathematica, and I just wasn't sure how to incorporate the surface BC. Is there a specific Finite Difference scheme to use at the boundary because let say I use central differencing. Won't that take me outside my boundary?

I'd like to see you're code if that's ok. Thanks again for the help.

hunt_mat

Essentially that is the thing I am saying, the only other thing is the BC on the boundary, so split your interval up into N pieces and you want to know how to compute your derivative on the boundary point [itex]x_{N}[/itex]. The wa yto go about this is to examine the point [itex]x_{N-\frac{1}{2}}[/itex].
The derivative is given by:
[tex]
\frac{dy}{dx}\Big|_{x_{N-\frac{1}{2}}}=\frac{y_{N}-y_{N-1}}{h}
[/tex]
Now the value of the derivative at [itex]N-1/2[/itex] is approximately the average of the derivatives at each side, so:
[tex]
\frac{dy}{dx}\Big|_{x_{N-\frac{1}{2}}}=\frac{1}{2}\left(\frac{dy}{dx}\Big|_{x_{N}}+\frac{dy}{dx} \Big|_{x_{N-1}}\right)
[/tex]
Then you use:
[tex]
\frac{dy}{dx}\Big|_{x_{N-1}}=\frac{y_{N}-y_{N-2}}{2h}
[/tex]
You solve for the thing you want
[tex]
\frac{dy}{dx}\Big|_{x_{N}}=\frac{3y_{N}-4y_{N-1}+y_{N-2}}{2h}
[/tex]

I have sent you my programs.

mykutu

Thanks for explaination. It helps me to solve manually.
Would you share with me matlab code. Because of i couldnt write.

hunt_mat

Check your mail on this site, I sent you the programs there.

mykutu

:( there isnt any new message in my mail or site private message inbox :( Would you send again

hunt_mat

I am at a conference currently and I will post it here so there is no mistakes.