The term transcritical flow describes the existence of both subcritical and supercritical flow simultaneously within the computational domain. Transcritical flow situations are encountered in many engineering problems of practical interest. Numerical simulation of such problems renders application of single-regime computational-codes awkward, at best, for steady flow, and impossible for unsteady flow. Therefore a generalized computational algorithm, that handles transcritical flow accurately and robustly, would find immediate applications. This study presents a second-order accurate (in both time and space) bidiagonal implicit scheme capable of resolving transcritical flow. The proposed scheme is of predictor-corrector type and has excellent shock-capturing capabilities due to an added adaptive dissipation term. Moreover, since the spatial discretization of the scheme involves only two computational points, hydraulic structures can be easily simulated. Overall the scheme is very simple to implement and holds promise for wide usage. The scheme has been extensively applied to a variety of problems of practical interest. Comparison of the computed solutions with other published results and/or exact solutions -only a small sample of which are included in this paper - clearly demonstrate its accuracy, robustness and shock-capturing abilities.