The numerical modeling of mechanical waves is currently a fundamental tool for the study and investigation of their propagation in media with heterogeneous physical properties and/or complex geometry, as, in these cases, analytical methods are usually not applicable. These techniques are used in geophysics (geophysical interpretation, subsoil imaging, development of new methods of exploration), seismology (study of earthquakes, regional and global seismology, accurate calculation of synthetic seismograms), in the development of new methods for ultrasonic diagnostics in materials science (non-destructive methods) and medicine (acoustic tomography). In this paper we present a review of numerical methods that have been developed and are currently used. In particular we review the key concepts and pioneering ideas behind finite-difference methods, pseudospectral methods, finite-volume methods, Galerkin continuous and discontinuous finite-element methods (classical or based on spectral interpolation), and still others such as physics-compatible, and multiscale methods. We focus on their formulations in time domain along with the main temporal discretization schemes. We present the theory and implementation for some of these methods. Moreover, their computational characteristics are evaluated in order to aid the choice of the method for each practical situation.

机械波的数值建模目前是研究和研究其在具有异质物理特性和/或复杂几何形状的介质中传播的基本工具，因为在这些情况下，分析方法通常不适用。这些技术用于地球物理学（地球物理解释，地下土壤成像，新的勘探方法的开发），地震学（地震研究，区域和全球地震学，合成地震图的精确计算）以及材料超声诊断新方法的开发。科学（非破坏性方法）和医学（声学层析成像）。在本文中，我们介绍了已开发并正在使用的数值方法。我们特别回顾了有限差分方法背后的关键概念和开创性思想，伪光谱方法，有限体积方法，Galerkin连续和不连续有限元方法（经典方法或基于光谱插值法），还有其他方法，例如物理兼容方法和多尺度方法。我们将重点放在时域上的公式以及主要的时间离散方案上。我们介绍了其中一些方法的理论和实现。此外，对它们的计算特性进行了评估，以帮助针对每种实际情况选择方法。我们介绍了其中一些方法的理论和实现。此外，对它们的计算特性进行了评估，以帮助针对每种实际情况选择方法。我们介绍了其中一些方法的理论和实现。此外，对它们的计算特性进行了评估，以帮助针对每种实际情况选择方法。