Numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both the forward modelling and migration of seismic wavefields in complicated geologic media. In P- and S-wave propagation, the finite-element method is a powerful tool for determining the effect of structural irregularities on wave propagation. Dependence of the wave equation on both spatial and temporal differentials requires solving both spatial and temporal discretization. In the spatial discretization step in 1D and 2D, piecewise linear basis functions and the Galerkin method are the most commonly used tools. After solving spatial discretization with the finite-element method, the wave equation reduces to an ODE (ordinary differential equation). In this regard, different authors used different ODE solver including Runge-Kutta method and finite-difference method. This paper will familiarize the reader with the diverse approaches of solving temporal discretization. An application of finite-element methods to solve seismic wave motion in linear viscoelastic media (where inelastic strains development depends not only on the current state of the stress and strain but on the full history of their development), using memory variable formalism in spatial discretization step, is one of the reviewed sections.
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