I would like to model heat diffusion at the gold / water interface after excitation of the metal surface by an ultrafast laser pulse (ca. 80 fs).
An appropriate model to start with would be the "two temperature model" (TTM) or "parabolic two step" (PTS). This model considers the electronic and lattice systems as separate with electronic and lattice temperature $T_e$ and $T_l$, respectively. The electronic and lattice systems are coupled through an electron-phonon coupling constant $G$ and the electronic system is excited by a source term $S$ representing the incoming laser pulse. The model can be written as two coupled PDEs [from Smith (2001) Appl. Phys. Lett. 78:1240]: $$ C_e(T_e) \frac{\partial T_e}{\partial t} = \frac{\partial}{\partial x} \left( k_e(T_e, T_l) \frac{\partial T_e}{\partial x} \right) - G \left[ T_e - T_l \right] + S \\ C_l \frac{\partial T_l}{\partial t} = G \left[ T_e - T_l \right], $$ where $C_e$, $C_l$ are electron and lattice heat capacities, respectively, and $k_e$ is the thermal conductivity.
I am specifically interested in determining the temperatures of the system for the first 100 ps.
The question I would like to ask is: What are the different methods and tools available to solve the problem, with an emphasis on entry-level methodologies?
