  A high-performance parallel algorithm is proposed for modeling the
propagation of acoustic and elastic waves in inhomogeneous media. An initial
boundary-value problem is replaced by a series of boundary-value problems for a
constant elliptic operator and different right-hand sides via the integral
Laguerre transform. It is proposed to solve difference equations by the
conjugate gradient method for acoustic equations and by the GMRES$(k)$ method
for modeling elastic waves. A preconditioning operator was the Laplace operator
that is inverted using the variable separation method. The novelty of the
proposed algorithm is using the Dichotomy Algorithm (Terekhov, 2010), which was
designed for solving a series of tridiagonal systems of linear equations, in
the context of the preconditioning operator inversion. Via considering
analytical solutions, it is shown that modeling wave processes for long
instants of time requires high-resolution meshes. The proposed parallel
fine-mesh algorithm enabled to solve real application seismic problems in
acceptable time and with high accuracy. By solving model problems, it is
demonstrated that the considered parallel algorithm possesses high performance
and efficiency over a wide range of the number of processors (from 2 to 8192).
