The numerical methods and algorithms for correct simulation of dynamic processes occurring in layered structures, composite materials, and geological media during seismic prospecting / seismic activity initiation are actively developed at the current moment. It should be noted that the ray-tracing methods widely used in the industry are approximate methods and do not make it possible to understand the behavior of most real fractured geological media. There are a great number of numerical methods for the simulation of seismic processes in complex media (the finite difference method, the Galerkin method, etc.) [1–3]. There are also various numerical methods actively used to construct hybrid calculation algorithms [4]. Also note that, as the defining system of equations of elasticity that describes the propagation of seismic waves is hyperbolic, its numerical solution can be carried out by the grid- characteristic method.

Apparently, the characteristic method was proposed for the first time in [5]. It was described in detail for a one-dimensional case in [6] and later generalized for a multidimensional case in [7]. Because the points at which the solution is calculated might become concentrated and, accordingly, the accuracy of the calculation could decrease, this method did not become widely used. The interpolation procedure was introduced in [8, 9], and it allowed converting this method into an inverse characteristic method (grid-characteristic method) [8, 9].

Earlier, the grid-characteristic method was used in the numerical solution of the problems of gas dynamics [10]. It was adapted to solve the problems of deformed solid mechanics only in 1980s [11, 12]. At first, seismic fields were described by the characteristic method on unstructured triangular grids. It was used in [13] for the numerical simulation of seismic response in multilayer geological media in a two-dimensional case. Later, the grid-characteristic method was generalized for the case of the presence of a fluid-saturated crack in a uniform elastic medium [14]. In [15], the numerical simulation of wave response was carried out with account for stratification and cracking. In [16], the response of a cluster (set) of fluid-saturated cracks was studied. Aside from the series of calculations in a two-dimensional case, a three-dimensional test calculation was carried out on unstructured grids. The modification of the method with the use of unstructured grids was described in detail in [17]. The main restriction in three-dimensional calculations is the computational complexity of the problem. However, if the grid-characteristic method proposed in [18] is used on hexahedral grids, then it is possible to increase the speed of calculations significantly and simulate the three-dimensional problem of the formation of the response of the cluster of fluid-saturated cracks located in a uniform medium.

In this report the results of the grid-characteristic method application in geological, seismic resistance estimation and railway non-destructing testing problems are presented (see https://www.scopus.com/authid/detail.uri?authorId=56290013800).

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