This paper provides an implicit fluctuation splitting scheme which achieves second-order accuracy in both space and time, while guaranteeing monotone solutions. The method is based on a dual-time-stepping approach which allows to separate the spatial and temporal discretizations. A novel distribution scheme of the unsteady-fluctuation term, coupled with a new limiting procedure for hampering spurious extrema, are the key-ingredients of the remarkable accuracy of the new method, which is applied with success to both scalar advection problems and unsteady inviscid flows in two space dimensions