  Numerical evolution of the spherically symmetric, massive Klein-Gordon field
is presented using a new adaptive mesh refinement (AMR) code with fourth order
discretization in space and time, along with compactification in space. The
system is non-interacting thus the initial disturbance is entirely radiated
away. The main aim is to simulate its propagation until it vanishes near
scri^+. By numerical investigations of the violation of the energy balance
relations, the space-time boundaries of ``well-behaving'' regions are
determined for different values of the AMR parameters. An important result is
that mesh refinement maintains precision in the central region for longer time
even if the mesh is only refined outside of this region. The speed of the
algorithm was also tested, in case of 10 refinement levels the algorithm was
two orders of magnitude faster than the extrapolated time of the corresponding
unigrid run.
