Over the past decades, the field of chemical engineering has witnessed an increased interest in unsteady-state processes. Multifunctional, as well as intensified chemical processes, may exhibit instationary behaviour especially when based on periodical operating conditions. Ideally, instationary processes lead to a higher yield and increased selectivities compared to con- ventional steady-state fixed-bed processes. Typical candidates among these are the reverse-flow-reactor, the chromatographic reactor and the adsorptive reactor. Since the underlying regeneration strategy is nearly always based on cycles   e.g. a reaction cycle is followed by a regeneration cycle and so on   the overall temporal behaviour of such processes eventually develops into cyclic steady states (after a transient phase). Experiments reveal a slow transient behaviour into the cyclic steady-state. This can also be observed in simulation based on conventional numerical treatment such as the method of lines. In addition to this problem many instationary processes exhibit sharp fronts or even shocks which require stabilisation of the convective terms. In this work we present a method of combining the idea of global discretisa- tion with modern stabilisation techniques of type FEM-FCT and FEM-TVD in order to obtain an e cient, well approximating and robust tool for the general simulation of instationary and in particularly cyclic-steady-state processes.