Meurig
Jan23-12, 09:07 AM
Hi all,
I am trying to construct a numerical solution to the following linear harmonic problem posed in a wedge of interior angle 0<\alpha<pi/2
\bigtriangledown^2\phi(r,\theta), \ r>0, \ -\alpha<\theta<0
\bigtriangledown\phi\cdot\mathbf{n}=0, r>0,\ \theta=-\alpha,
\frac{\pi}{\alpha}\eta(r)-2r\eta_{r}-\frac{1}{r}\phi_{\theta}=0, r>0, \theta=0,
(1+\frac{\pi}{\alpha})\phi - 2r\phi_{r}+(1+\sigma\tan(\alpha))\eta =0, r>0, \theta=0,
In addition I have the far field boundary conditions:
\phi=r^{\frac{\pi}{2\alpha}}\sin(\frac{\pi\theta}{ 2\alpha}) as r\rightarrow\inf
\eta=\frac{\pi}{4\alpha}r^{\frac{\pi}{2\alpha}-1} as r\rightarrow\inf.
And the solution local to the tip of the wedge given by
\phi=\frac{A\alpha\sin{\alpha}(1+\sigma\tan{\alpha })}{\pi(1+\pi/\alpha)}+rA\cos(\theta+\alpha)
\eta=-\frac{A\alpha\sin\alpha}{\pi}+\eta_1 r
where A and \eta_1 can be approximated through solving the near field boundary condition
\phi_\theta +r\tan(\theta+\alpha)\phi_r=0, r=\epsilon, -\alpha<\theta<0
So far I have attempted constructing a finite difference approximation in terms of polar coordinates, but as I iterate this scheme the error increases exponentially until phi approaches infinity.
I wonder if anyone has any ideas with regards to what I should be looking to do/what I should be weary of.
Cheers,
Meurig
*edit to correct latex
I am trying to construct a numerical solution to the following linear harmonic problem posed in a wedge of interior angle 0<\alpha<pi/2
\bigtriangledown^2\phi(r,\theta), \ r>0, \ -\alpha<\theta<0
\bigtriangledown\phi\cdot\mathbf{n}=0, r>0,\ \theta=-\alpha,
\frac{\pi}{\alpha}\eta(r)-2r\eta_{r}-\frac{1}{r}\phi_{\theta}=0, r>0, \theta=0,
(1+\frac{\pi}{\alpha})\phi - 2r\phi_{r}+(1+\sigma\tan(\alpha))\eta =0, r>0, \theta=0,
In addition I have the far field boundary conditions:
\phi=r^{\frac{\pi}{2\alpha}}\sin(\frac{\pi\theta}{ 2\alpha}) as r\rightarrow\inf
\eta=\frac{\pi}{4\alpha}r^{\frac{\pi}{2\alpha}-1} as r\rightarrow\inf.
And the solution local to the tip of the wedge given by
\phi=\frac{A\alpha\sin{\alpha}(1+\sigma\tan{\alpha })}{\pi(1+\pi/\alpha)}+rA\cos(\theta+\alpha)
\eta=-\frac{A\alpha\sin\alpha}{\pi}+\eta_1 r
where A and \eta_1 can be approximated through solving the near field boundary condition
\phi_\theta +r\tan(\theta+\alpha)\phi_r=0, r=\epsilon, -\alpha<\theta<0
So far I have attempted constructing a finite difference approximation in terms of polar coordinates, but as I iterate this scheme the error increases exponentially until phi approaches infinity.
I wonder if anyone has any ideas with regards to what I should be looking to do/what I should be weary of.
Cheers,
Meurig
*edit to correct latex