In the present manuscript we consider the problem of dispersive wave simulation on a rotating globally spherical geometry. In this Part IV we focus on numerical aspects while the model derivation was described in Part III. The algorithm we propose is based on the splitting approach. Namely, equations are decomposed on a uniform elliptic equation for the dispersive pressure component and a hyperbolic part of shallow water equations (on a sphere) with source terms. This algorithm is implemented as a two-step predictor-corrector scheme. On every step we solve separately elliptic and hyperbolic problems. Then, the performance of this algorithm is illustrated on model idealized situations with even bottom, where we estimate the influence of sphericity and rotation effects on dispersive wave propagation. The dispersive effects are quantified depending on the propagation distance over the sphere and on the linear extent of generation region. Finally, the numerical method is applied to a couple of real-world events. Namely, we undertake simulations of the BULGARIAN 2007 and CHILEAN 2010 tsunamis. Whenever the data is available, our computational results are confronted with real measurements.