  This paper discusses the technical aspects - mathematical and numerical -
associated with the numerical simulations of a mesoscopic system in the time
domain (i.e. beyond the single frequency AC limit). After a short review of the
state of the art, we develop a theoretical framework for the calculation of
time resolved observables in a general multiterminal system subject to an
arbitrary time dependent perturbation (oscillating electrostatic gates, voltage
pulses, time-vaying magnetic fields) The approach is mathematically equivalent
to (i) the time dependent scattering formalism, (ii) the time resolved Non
Equilibrium Green Function (NEGF) formalism and (iii) the partition-free
approach. The central object of our theory is a wave function that obeys a
simple Schrodinger equation with an additional source term that accounts for
the electrons injected from the electrodes. The time resolved observables
(current, density. . .) and the (inelastic) scattering matrix are simply
expressed in term of this wave function. We use our approach to develop a
numerical technique for simulating time resolved quantum transport. We find
that the use of this wave function is advantageous for numerical simulations
resulting in a speed up of many orders of magnitude with respect to the direct
integration of NEGF equations. Our technique allows one to simulate realistic
situations beyond simple models, a subject that was until now beyond the
simulation capabilities of available approaches.
