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 and time) that completely avoids inner iterations. Numerical examples are presented.

翻译：本文提出了一种针对可能含振荡系数的声波方程的离散化方法，该方法基于空间局部子问题的离散解的叠加，并采用隐式时间离散化进行计算。通过利用全局系统矩阵元素的指数衰减性质及适当的单位分解，证明了局部解的叠加与（全局）隐式格式的解足够接近，从而合理化了在多个重叠子域上进行局部化（尤其是并行）计算。此外，为保持子域间适度的重叠，引入了固定时间步数后的重启机制。总体上，该方法可视为一种完全避免内部迭代的空间-时间域分解策略。文中给出了数值算例。