While the Gross--Pitaevskii equation is well-established as the canonical dynamical description of atomic Bose-Einstein condensates (BECs) at zero-temperature, describing the dynamics of BECs at finite temperatures remains a difficult theoretical problem, particularly when considering low-temperature, non-equilibrium systems in which depletion of the condensate occurs dynamically as a result of external driving. In this paper, we describe a fully time-dependent numerical implementation of a second-order, number-conserving description of finite-temperature BEC dynamics. This description consists of equations of motion describing the coupled dynamics of the condensate and non-condensate fractions in a self-consistent manner, and is ideally suited for the study of low-temperature, non-equilibrium, driven systems. The delta-kicked-rotor BEC provides a prototypical example of such a system, and we demonstrate the efficacy of our numerical implementation by investigating its dynamics at finite temperature. We demonstrate that the qualitative features of the system dynamics at zero temperature are generally preserved at finite temperatures, and predict a quantitative finite-temperature shift of resonance frequencies which would be relevant for, and could be verified by, future experiments.
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