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I am looking for a rigorous way to deal with discontinuous thermal conductivity $K(x)$ in the heat equation (e.g. on the boundary of two different materials)

$$ u' = \nabla_x\cdot (K(x)\nabla_x u) + \text{\{possible other terms\}},$$

where $u(t,x)$ is the temperature. Clearly the equation breaks down at discontinuity since the derivative of $K$ does not exist, what becomes the boundary condition of the non-discretized equation above to replace the discontinuity?

When the equation is discretized, for basic finite difference scheme at boundary it is possible to in a loose sense use the "average value" (is there a derivation somewhere?) $$K(x)\approx (1/2) (K(x+1)+K(x)).$$

For finite volume scheme not at the boundary $$Au' = \int\int\int (\nabla_x\cdot K\nabla_x u)dA$$ $$=\int\int (K\nabla_x u) \cdot \bar ndS$$ $$=K\int \int\nabla_xu \cdot \bar n dS$$ Let's say a finite difference scheme is applicable, although there could be more generality $$\approx K \sum_k (u(x_1,...,x_k+1,...,x_d) + u(x,...,x_k-1,...,x_d)-2u(x))$$

The average value approximation then yields $$\approx \sum_k \frac{1}{2}(K(x)+K(x_1,...,x_k+1,...,x_d)) \cdot (u(x_1,...,x_k+1,...,x_d) -u(x)) +...,$$ for the finite volume method, is it valid on the boundary? References to books are welcome.

Qmechanic
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Magemathician
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    The first who studied and found solutions for discontinuities in the heat equation were John Forbes Nash and Ennio De Giorgihttps : https://www.karlin.mff.cuni.cz/~kaplicky/pages/pages/2011z/Nash1958.pdf – The Tiler Jun 15 '23 at 08:49
  • @HVAC I had no idea Nash worked in this area of mathematics... Anyway, since the equation is not defined at the boundary should not the boundary condition be defined rather than derived? – Magemathician Jun 15 '23 at 10:39
  • Related, if not dupe of, https://physics.stackexchange.com/q/107761/25301 – Kyle Kanos Jun 15 '23 at 11:42
  • @KyleKanos It is referenced in the question. Particularly, whether the same "averaging" works on finite volume is being asked. – Magemathician Jun 15 '23 at 11:46
  • Magemathician: Sorry, I'm not a specialist in the field, However, the first notable use of his work was the solution of Hilbert's 19th problem on the analyticity of minimizers of functionals with analytic integrand. – The Tiler Jun 15 '23 at 13:23
  • In any case, we use the equation for a multi-layered wall... – The Tiler Jun 15 '23 at 13:31
  • @HVAC In fact there is no boundary condition, and the flux is assumed differentiable despite the discontinuity in $K$. – Magemathician Jun 16 '23 at 09:31

2 Answers2

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Put a grid point $x_j$ at the interface between the two regions. Then $$k^+\frac{T_{j+1}-T_j}{\Delta x}=k^-\frac{T_j-T_{j-1}}{\Delta x}$$That is, the heat flux is continuous at the interface.

Chet Miller
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The scheme appears applicable, as discussed here. First, from Chet Miller's answer and comments the divergence theorem must hold and the flux be cotinuously differentiable, so at boundary the quantity of interest is $$\int\int (K(x)\nabla_x u) \cdot \bar ndS.$$ Now make the approximation the quantities vary linearly within the cell and there are finite $N$ number of faces. Apparently there is a theorem that then the integral is equal to the value found at the face centroid $$\approx \sum_{f=1}^N K_f (\nabla u)_f \cdot \bar n_f S_f,$$

where the subscript denotes the value at the face centre (and $S_f$ the surface area of the face). To estimate the face-value central differencing can be used for $K_f$, which gives the weighted average, weighted by the relative distance to the face.

Magemathician
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