I am attempting to create a finite difference solution for 1D Heat Diffusion through a composite wall with an abrupt material change and no thermal boundary resistance, assuming temperature- and position-dependent (due to material change) thermophysical properties. It should be noted that the two sides of the composite wall are held at a constant temperature.
The following differential equation is essentially what I need to discretize. $$\frac{\partial}{\partial t}\left[T(x)\right] = \frac{\partial}{\partial x}\left[\alpha \left[x,T(x)\right] \frac{\partial}{\partial x}\left[T(x)\right]\right]$$ Expanding thermal diffusivity $\alpha$ in terms of its subcomponents: $$\frac{\partial}{\partial t}\left[T(x)\right] = \frac{\partial}{\partial x}\left[\frac{k \left[x, T(x)\right]}{\rho(x) \cdot c_{P}\left[x, T(x)\right]} \cdot \frac{\partial}{\partial x}\left[T(x)\right]\right]$$ This assumes that density is solely dependent on position, not temperature. I have interpolated functions for $c_P$ and $k$, and hence have an interpolated $\alpha$. These can be seen below.
Density (rho) Isobaric Specific Heat (c_P) Thermal Conductivity (k) Thermal Diffusivity (alpha)
I'm a little confused about how I would discretize this. For the simpler, constant $\alpha$ case, we can discretize
$$\frac{\partial T}{\partial t} = \frac{\partial}{\partial x} \left[\alpha \frac{\partial T}{\partial x} \right]$$
By bringing out the alpha and just doing a second discrete differential:
$$\frac{1}{\alpha}\frac{T_m^{p+1}-T_m^{p}}{\Delta t} = \frac{T_{m+1}^p+T_{m-1}^p-2T_m^p}{(\Delta x)^2}$$