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I am trying to solve the 2-d heat equation on a rectangle using finite difference method. I am confused as to how to incorporate non linear radiation boundary condition at the edge.

$-k\frac{\partial T}{\partial n}=\sigma\epsilon(T^4-T^4_{\infty})+h(T-T_{\infty})$

Is the following implementation reasonable?

$-k\frac{T^{n+1}_{i}-T^{n+1}_{i-1}}{\Delta x}=\sigma\epsilon((T^n_{i})^4-T^4_{\infty})+h(T^{n+1}_{i}-T_{\infty})$.

My logic is that taking sufficiently small time steps will give accurate results using this formulation.

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  • $\begingroup$ Your logic is fine but probably you have typos here cause first the correct radiative BC is: $$-k \frac{\partial T}{\partial \vec{n}} = \sigma \epsilon (T^{4} - T_{\infty}^{4})$$ and you discretize it by finite difference to get: $$-k \frac{T_{i}^{n} - T_{i-1}^{n}}{\Delta x} = \sigma \epsilon ((T_{i}^{n})^{4}-T_{\infty}^{4})$$ $\endgroup$
    – Mithridates the Great
    Feb 17, 2020 at 19:23
  • $\begingroup$ Thanks for pointing out the mistake !! I have edited my question. Anyways, I did take the corrections into account in my code. Running the actual simulations gives a higher boundary temperature value then just using convective boundary condition which should not happen physically. I am not sure why does this happen. $\endgroup$
    – Nitin
    Feb 18, 2020 at 5:30
  • $\begingroup$ Sorry, but still you have the same problem in your edit. Please pay attention to the superscripts. In the first term of right hand side $n$ should be $n+1$ as well. $\endgroup$
    – Mithridates the Great
    Feb 18, 2020 at 17:10
  • $\begingroup$ Are you using an explicit or implicit time-marching scheme? $\endgroup$
    – nicoguaro ♦
    Feb 19, 2020 at 11:50
  • $\begingroup$ I am using an explicit time marching scheme. $\endgroup$
    – Nitin
    Feb 20, 2020 at 12:10

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