We report on the development of a fully time-dependent computer simulation of the 4He normal fluid/superfluid (He-I/He-II) interface dynamics. Both the diverging thermal conductivity and the rapidly changing heat capacity of 4He near the lamda point present a challenge to traditional numerical methods for integrating the heat diffusion equation. We use the Dupont-II three-time-level method known for its good convergence properties for strong nonlinear models. Still, this algorithm does not conserve energy precisely, so a supplementary convergence criterion based on absolute enthalpy error is developed. We report on the underlying theoretical model, the numerical algorithm and its implementation, and simulation results. We compare the results with experimental data and discuss the error analysis and the balance between simulation fidelity and convergence.