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Suppose I have to solve the 2-D heat equation in a rectangular domain using the finite difference method, for the boundary conditions say:

$T_1$ is the temperature of the right side of the rectangle,

$T_2$ is the temperature of the top side,

$T_3$ is the temperature of the left side, and $T_4$ is the temperature at the bottom.

           T2
    _______________
   |               |
T3 |               |T1
   |               |
   |_______________|
          T4

at the vertices points of the rectangle, how to set the boundary conditions? do I need to take the average value of the two temperatures?

Regards

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  • $\begingroup$ For your finite difference, if you are you doing something like $\nabla^2 u = \frac{u_{i+1,j}-2u_{i,j} + u_{i-1,j}}{\Delta x^2} + \frac{u_{i,j+1}-2u_{i,j} + u_{i,j-1}}{\Delta y^2}$ then the corner points wont affect the solution on the interior. $\endgroup$ – Steve May 24 '16 at 14:38
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I believe you cannot apply two boundary conditions at the same point. You need to choose between T2 and T3 for the top-left point for instance. The average might be acceptable but I'm not sure of the physical meaning though.

Anyway, with a fine discretization, it should not be a problem since the consequence of this choice will be insignificant on the solution.

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  • $\begingroup$ Thank you I got the idea, physically the vertices must have a determined temperature $\endgroup$ – Navaro May 24 '16 at 14:33
  • $\begingroup$ @Navaro, but the question is, does your physical arrangement expect a jump at that point or the average or something else? You can model basically any of these by how you set the value at this point on your mesh. $\endgroup$ – Bill Barth May 24 '16 at 16:52
  • $\begingroup$ @Bill: I think it will give a similar results $\endgroup$ – Navaro May 24 '16 at 20:00
  • $\begingroup$ @Navaro, but you still have to pick something. Better to pick something closest to the physics you want to model, or mathematically more tractable. As you refine the mesh, your choice may make the problem harder if you try to model a jump rather than if you pick the average. Does this matter to you? Can you prove that the results will be similar? $\endgroup$ – Bill Barth May 24 '16 at 23:10
  • $\begingroup$ @Bill Physcally how to impose the Top edge at $T_2$ and the right edge to $T_1$ , does that really make sense? of course not $\endgroup$ – Navaro May 24 '16 at 23:20

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