I am consider the following free boundary problem: $u_t + \mathcal{L}u = 0, 0<x<s(t)$, with the boundary conditions: $u(T,x) = f_T(x)$, $u(t,s(t)) = g(t)$ and $u_x(t,s(t))=C$. Here, $\mathcal{L}$ is a linear differential operator.
I am planning to use the finite difference scheme to solve the problem. However, how should I update the free boundary when using FD? Any reference will be appreciated.