We deal with an initial-boundary value problem for the multidimensional acoustic wave equation, with the variable speed of sound. For a three-level semi-explicit in time higher-order vector compact scheme, we prove stability and derive 4th order error bound in the enlarged energy norm. This scheme is three-point in each spatial direction, and it exploits additional sought functions which approximate 2nd order non-mixed spatial derivatives of the solution to the equation. At the first time level, a similar two-level in time scheme is applied, with no derivatives of the data. No iterations are required to implement the scheme. We also present results of various 3D numerical experiments that demonstrate a very high accuracy of the scheme for smooth data, its advantages in the error behavior over the classical explicit 2nd order scheme for nonsmooth data as well and an example of the wave in a layered medium initiated by the Ricker-type wavelet source function.

翻译：本文研究具有变声速的多维声波方程初边值问题。针对一种三层半显式时间高阶向量紧致格式，我们证明了其稳定性，并导出了在扩充能量范数下的四阶误差界。该格式在每个空间方向采用三点模板，并引入附加待求函数来逼近方程解的二阶非混合空间导数。在第一时间层，应用了一种无需数据导数的类似两层时间格式。该格式实现无需迭代。我们还展示了各种三维数值实验的结果：对于光滑数据，该格式展现出极高的精度；对于非光滑数据，其在误差行为上优于经典显式二阶格式；同时给出了由Ricker型子波源函数引发的层状介质中波的传播实例。