The flux-corrected transport (FCT) technique and the alternating-direction explicit (ADE) method are coupled through a time-splitting technique. This new combination of both methods has been used successfully to solve the fully coupled Navier–Stokes system applied to ionospheric thermal plasma flows with a viscosity and strong heat conduction. The combined scheme gives convergent solutions within the time step set for nonlinear stability of the corresponding nondissipative flow fields, and the time-dependent solutions are consistent with other model results using different methods. To have a quantitative view of the flux-limiter of Boris' FCT version, a concept of local variation is defined to identify local extrema. The total variation diminishing scheme finds unique entropy solutions for vanishing dissipation. The ADE scheme, however, enables us to handle dissipation when the FCT technique alone can be inappropriate.