I am assosiated on a research project, where we wish to simulate a physical system with a system of heat equations as the governing equations. An example in $1D$ would look something like the following:
\begin{align} \frac{1}{\alpha_1} \frac{\partial u_1}{\partial t} &= \frac{\partial^2 u_1}{\partial x^2} + P_0(1+\cos(2\omega t)), \quad x\in [-L,0] \\\ \frac{1}{\alpha_2} \frac{\partial u_2}{\partial t} &= \frac{\partial^2 u_2}{\partial x^2}, \quad x\in [0,L], \end{align}
where $\alpha_1\neq \alpha_2$, $\frac{\partial u_1(-L,t)}{\partial x} = 0 = \frac{\partial u_2(L,t)}{\partial x}$, $u_1(x,0)=f_1(x)$ and $u_2(x,0)=f_2(x)$. Furthermore, we might assume $f_1$ and $f_2$ are constant. At the boundary between bodies, i.e. $x=0$, we assume there is a continuous flow of heat, and this is where my problem comes in.
It is easy enough to solve the heat equations seperately using standard finite difference methods, but how do I deal with this shared boundary where there is a continuous flow? I have tried to find literature, code examples, etc., which might help me understand this but came up empty handed.
Thank you in advance.