Vasiliev's fully implicit finite difference box scheme, used in hydraulic engineering to simulate flood routing and overland flow, is invalid, in its usual implementation, for modelling transcritical flow. Short waves, generated near regions of steep gradients, and responsible for the often observed grid-to-grid oscillations, are not damped. In order, to suppress these spurious oscillations, which degrade and successively instabilize the solution, the adaptive smoothing approach is applied herein. The remedy confines the damping only near the jump of a discontinuity, sharp gradients, and unaffects the regions where the flow is relatively smooth. In addition, weighted in time the spatial derivative approximations, the prevention of the dissipation in a broad spectrum of wave number, is obtained. Consistency and accuracy, are analyzed. By Fourier linear analysis, the stability, the convergence, and the variations of the time-weighted parameter, of the Courant number, of the Froude number, of the frictional parameter, and of the viscosity coefficient are investigated. In addition, for the role of the viscosity coefficienty on dissipation and dispersion comparisons with the asymptotic analysis, are presented. The exact conservation property is theoretically, and numerically verified. Benchmark test cases, involving transcritical flow, friction, nonuniform bed slopes, and nonprismatic channels, and laboratory dam-break simulations, compared to the analytical solutions, and the experimental data, are presented.