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.

翻译：本文提出了一种基于隐式时间离散化的空间局部子问题离散解叠加来离散化具有振荡系数的声波方程的方法。基于全局系统矩阵的指数衰减项和合适的单位分解，证明了局部解的叠加与（全局）隐式格式的解足够接近，从而验证了在多个重叠子域上进行局部（尤其是并行）计算的合理性。此外，通过在一定时间步数后引入重启动策略来保持子域间的适度重叠。整体而言，该方法可理解为在连续短时间区间内完全避免内部迭代的空间域分解策略。文中给出了数值算例。