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Numerical evolution of the spherically symmetric, massive Klein-Gordon fieldis presented using a new adaptive mesh refinement (AMR) code with fourth orderdiscretization in space and time, along with compactification in space. Thesystem is non-interacting thus the initial disturbance is entirely radiatedaway. The main aim is to simulate its propagation until it vanishes nearscri^+. By numerical investigations of the violation of the energy balancerelations, the space-time boundaries of ``well-behaving'' regions aredetermined for different values of the AMR parameters. An important result isthat mesh refinement maintains precision in the central region for longer timeeven if the mesh is only refined outside of this region. The speed of thealgorithm was also tested, in case of 10 refinement levels the algorithm wastwo orders of magnitude faster than the extrapolated time of the correspondingunigrid run.Comment: 12 pages, 7 figures; major conceptual changes, other tests added;  accepted for publication by Int.J.Mod.Phys.

TESTING A NEW MESH REFINEMENT CODE IN THE EVOLUTION OF A SPHERICALLY SYMMETRIC KLEIN–GORDON FIELD