  This work proposes a discretization of the acoustic wave equation with
possibly oscillatory coefficients based on a superposition of discrete
solutions to spatially localized subproblems computed with an implicit time
discretization. Based on exponentially decaying entries of the global system
matrices and an appropriate partition of unity, it is proved that the
superposition of localized solutions is appropriately close to the solution of
the (global) implicit scheme. It is thereby justified that the localized (and
especially parallel) computation on multiple overlapping subdomains is
reasonable. Moreover, a re-start is introduced after a certain amount of time
steps to maintain a moderate overlap of the subdomains. Overall, the approach
may be understood as a domain decomposition strategy in space on successive
short time intervals that completely avoids inner iterations. Numerical
examples are presented.
