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A new numerical method is introduced to study the problem of time evolutionof generic non-linear dynamical systems in four-dimensional spacetimes. It isassumed that the time level surfaces are foliated by a one-parameter family ofcodimension two compact surfaces with no boundary and which are conformal to aRiemannian manifold C. The method is based on the use of a multipole expansiondetermined uniquely by the induced metric structure on C. The approach is fullyspectral in the angular directions. The dynamics in the complementary 1+1Lorentzian spacetime is followed by making use of a fourth order finitedifferencing scheme with adaptive mesh refinement.  In checking the reliability of the introduced new method the evolution of amassless scalar field on a fixed Kerr spacetime is investigated. In particular,the angular distribution of the evolving field in to be superradiant scatteringis studied. The primary aim was to check the validity of some of the recentarguments claiming that the Penrose process, or its field theoreticalcorrespondence---superradiance---does play crucial role in jet formation inblack hole spacetimes while matter accretes onto the central object. Ourfindings appear to be on contrary to these claims as the angular dependence ofa to be superradiant scattering of a massless scalar field does not show anypreference of the axis of rotation. In addition, the process of superradiance,in case of a massless scalar field, was also investigated. On contrary to thegeneral expectations no energy extraction from black hole was found even thoughthe incident wave packets was fine tuned to be maximally superradiant. Insteadof energy extraction the to be superradiant part of the incident wave packetfails to reach the ergoregion rather it suffers a total reflection whichappears to be a new phenomenon.

On the Use of Multipole Expansion in Time Evolution of Non-linear Dynamical Systems and Some Surprises Related to Superradiance