One of the main issues that has to be faced when evaluating lightning-induced overvoltages on a realistic network configuration is to implement the algorithm that solves the field-to-line coupling problem into an electromagnetic simulator that is able to represent with a high level of detail all the power system components. On one hand, numerical stability conditions of the finite-difference time-domain schemes have been deeply investigated in the presence of boundary conditions known analytically; on the other hand, there is not so much literature on the situations in which such conditions are provided in a numerical way through an external electromagnetic simulator. The study reveals that the commonly used linear extrapolation of currents at the line extremities might rise to numerical instabilities. An efficient solution using the method of characteristics is presented and validated.