Question-and-Answer Resource for the Building Energy Modeling Community
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Hi, very interesting question. The answer is found in HeatBalFiniteDifferenceManager.cc since it appears to not be documented elsewhere (see e.g. line 1934 onward in the IntInterfaceNodeEqns function).
At a node separating 2 massless materials, $C_p$ and $\rho$ are 0 and simplifying the implicit heat equation in your post yields
$T_i^{j+1} = \frac{R_{i-1} T_{i+1}^{j+1} + R_{i+1} T_{i-1}^{j+1}}{R_{i+1}+R_{i-1}}$,
where $T_i^{j+1}$ is the temperature at node $i$ at the newest timestep $j + 1$, and $R$ is thermal resistance. Transient behavior at node $i$ may be driven by transient boundary conditions or propagate from non-massless materials in the model. This is the equation used by EnergyPlus in this situation (line 1942 in the code).
When a node separates a massless material (e.g. on the left at $i-1$) and a non-massless material (e.g. on the right at $i+1$), $C_p$ and $\rho$ are retained for the non-massless material in the implicit heat equation. However, the $\frac{k_E}{\Delta x}$ term corresponding to the massless material is related to the thermal resistance of the massless material $R_{i-1}$ and essentially replaced by $\frac{1}{R_{i-1}}$ (line 2080 in the code).
