The paper presents an approach to the numerical simulation of sonic logging for arbitrary Tilted Transversely Isotropic (TTI) heterogeneous surrounding media. The simulation is based on the Lebedev's finite difference scheme. In order to take into account the sharpest interface of the problem, namely the interface between the fluid-filled borehole and the surrounding media, the analysis and simulation are performed in a cylindrical coordinate system. The use of this particular coordinate system causes two mathematical peculiarities: azimuth inflation of the mesh cells and singularity at the vertical axis r = 0. To solve these problems, local azimuth mesh refinement and interpolation at r = 0 are used, which are presented and verified in the paper. Another important factor in constructing an effcient and reliable method for the numerical simulation of the sonic log is the limitation of the computational domain. The use of the standard PML method for anisotropic media often leads to numerical instability. Therefore, its modification, known as MPML, is used here. The numerical simulation of acoustic wave fields in realistic borehole models requires the use of high performance computing systems with parallel architecture. For this purpose, we apply the spatial decomposition of the computational domain and organise the computations using MPI. The results of numerical experiments are presented and discussed.