Abstract <p>Nowadays, the solution of applied problems of seismic exploration and ultrasonic nondestructive testing is accompanied by the use of computer simulation. This poses a challenge for scientists to develop new modifications of numerical methods that can increase calculation accuracy while minimizing the cost of computing resources. In contrast to numerical methods on unstructured computational meshes, the use of Chimera computational meshes (or superimposed or adaptive meshes) makes it possible to describe boundaries and contact boundaries of arbitrary shape, but at the same time spending less RAM and time on calculations. This is especially important for the active use of neural networks in solving inverse problems, since when training samples are generated, the accuracy of modeling, speed of calculations, and amount of RAM spent are crucial. The paper considers and compares various modifications of the grid-characteristic method on Chimera computational meshes. Examples of test calculations are given.</p>