thermodynamics - How to solve the heat equation for compound materials with different heat conductivities numerically?

I'm solving the heat equation with time dependent boundary conditions numerically in a 2D system using the ADI scheme. For the purpose of this question, let's assume a constant heat conductivity and assume a 1D system, so

$$\rho c_p \frac{\partial T}{\partial t} = \lambda \frac{\partial^2 T}{\partial x^2}.$$ This works very well, but now I'm trying to introduce a second material. This one differs slightly in heat capacity and density but has a very different heat conductivity and is connected to the other material by a sharp interface, i.e. a stepwise change in $\lambda$.

How should this be treated numerically in the ADI scheme? I can think of different approaches:

1. Treat the two materials as independent domains and connect them by a boundary condition calculating the heat flow in and out of the interface in terms of temperature on the other side of the interface in the last time step. Use a simple forward difference for that on both sides of the interface.


2. Treat it as one domain and use a very fine discretization close the interface as compared to the homogeneous material. Use a scheme like $$\lambda_{left} \frac{T_i - T_{i-1}}{\Delta x_i} = \lambda_{right} \frac{T_j - T_{j+1}}{\Delta x_j},$$ where $i$ and $j$ are the points left and right of the interface, instead of the the standard ADI for those points.


3. Drop the assumption of constant heat conductivity and use $$\rho c_p \frac{\partial T}{\partial t} = \frac{\partial \lambda}{\partial x}\frac{\partial T}{\partial x} + \lambda \frac{\partial^2 T}{\partial x^2}.$$ But in order do so one needs to approximate derivative of lambda at the step position, i.e. introduce an unknown characteristic width $s$ of the sharp interface. I assume, that the (more or less arbitrary) choice of this width will significantly influence the system's behaviour.

Any advice?