Terms number determination at the series truncation for the numerical solution of the problem of acoustic scattering from a sound-permeable spheres set

Acoustic scattering from small-sized obstacles under external influence is one of the most important problems in acoustics, primarily because of the practical applications of this phenomenon. The solution of this problem is reduced to solving the Helmholtz equation for a complex potential with certain boundary conditions. When using the calculation method based on the fast multipole method, the potentials are decomposed into series according to special spherical functions, the form of which depends on the region in which this potential is calculated. As a result, the numerical implementation of the resulting matrix system raises the question of the correct choice of the number of series members when truncating them, since with a small number of series members, the calculation accuracy will be low, and with a large one will be increase not only the accuracy, but also the calculation time. An analysis of the scientific literature has shown that there are two approaches to choosing the number of terms of a series when truncating for such problems. In the first approach, truncation of the series is based on comparing two consecutive values of the sum of the sought series until the required degree of accuracy is achieved. In the second approach, all series in each expansion are truncated for a fixed number of series terms determined using heuristic formulas. In this paper, using the example of three sound-permeable spheres of different radii in the case of their strong interaction, when numerical calculations become «sensitive» to the number of terms during truncation, we compared these approaches. The analysis of the obtained data showed that to determine the value of the desired function with the necessary accuracy, it is optimal to use a combination of the considered approaches.