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For a project I am assigned to solve the heat equation in a 2D environment in Python. To do this, I am using the Crank-Nicolson ADI scheme and so far things have been going smooth. I would also like to add that this is the first time that I have done numerical computing like this and I don't have a lot of experience with PDE's and finite difference methods.

However, problems start when I try to add a second material with a different thermal diffusivity. At the boundaries of the 2 materials, the temperature keeps rising and rising exponentially, but only at the boundaries (as if they are a heat source). Any other place in the domain functions fine and does not show this behaviour at all. Why does this happen and how can I stop it? Do I need to impose new boundary conditions on these boundaries, or can I just 'force' my way through this by calculating the temperature on the boundary and treating the two materials as independent domains?

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    $\begingroup$ What is your finite difference equation for matching the heat fluxes at the boundary? $\endgroup$
    – Chet Miller
    Commented Apr 12, 2018 at 13:43
  • $\begingroup$ Currently the whole domain has fixed temperature boundary conditions at the edges of the (square) domain. I haven't implemented boundary conditions between the two materials because I thought the program could just calculate the temperature values, only with a different thermal diffusivity. I know there is such a thing as thermal resistance, but I wanted to add that later. I just cant understand why the boundaries between the materials get so 'hot'. $\endgroup$
    – Jeroen Reurink
    Commented Apr 12, 2018 at 13:59
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    $\begingroup$ Like I said, what is the heat flux matching boundary condition between the two materials? $\endgroup$
    – Chet Miller
    Commented Apr 12, 2018 at 14:38
  • $\begingroup$ Related: physics.stackexchange.com/q/107761/25301, physics.stackexchange.com/q/13996/25301, physics.stackexchange.com/q/137705/25301, etc $\endgroup$
    – Kyle Kanos
    Commented Apr 12, 2018 at 18:08
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    $\begingroup$ I'm voting to close this question as off-topic because it is about debugging code and not about a physics concept. $\endgroup$
    – BioPhysicist
    Commented Dec 18, 2019 at 19:03

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The reason it got 'hot' is the interface between the two parts is not properly set. you cannot set a fixed temperature boundary condition on both parts. This "temperature" is calculated but not given. If you set a "guessed" temperature, the local temperature gradient will be unphysical. On one part, it will be a heat source or sink.

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