We consider an initial-boundary value problem for the $n$-dimensional wave equation with the variable sound speed, $n\geq 1$. We construct three-level implicit in time and compact in space (three-point in each space direction) 4th order finite-difference schemes on the uniform rectangular meshes including their one-parameter (for $n=2$) and three-parameter (for $n=3$) families. We also show that some already known methods can be converted into such schemes. In a unified manner, we prove the conditional stability of schemes in the strong and weak energy norms together with the 4th order error estimate under natural conditions on the time step. We also transform an unconditionally stable 4th order two-level scheme suggested for $n=2$ to the three-level form, extend it for any $n\geq 1$ and prove its stability. We also give an example of a compact scheme for non-uniform in space and time rectangular meshes. We suggest simple fast iterative methods based on FFT to implement the schemes. A new effective initial guess to start iterations is given too. We also present promising results of numerical experiments.

翻译：我们考虑的是美元-维波方程式的初步界限值问题,使用可变的音速($n\geq 1美元)。我们用空间时间和紧凑度(每个空间方向三点)构建了在时间和空间(每个空间方向三点)中隐含的三种等级的隐含时间和紧凑度(每个空间方向三点),在统一的矩形间螺旋(包括它们的单数)中,包括它们的单数(n=2美元)和三分(n=3美元)家庭,我们同时展示了某些已知的方法可以转换成这种计划。我们以统一的方式证明了强弱能源规范中的计划有条件的稳定性,以及在时间步骤的自然条件下的第四级误差估计。我们还将一个无条件稳定的第四级两级计划($=2美元)转化为三级计划,将其扩展至任何一美元/美元,并证明其稳定性。我们还举了一个在空间和时间(n=3美元)中非统一型的契约计划的例子。我们建议基于FFT的简单快速迭接合方法来实施这些计划。我们还给出了启动这种实验的新的有效初步猜测,现在也提出了有希望的数字。