Investigation of the effectiveness of numerical methods for solving a mathematical model of high-frequency geoacoustic emission

This paper presents a study of four numerical methods for solving a mathematical model of high-frequency geoacoustic emission: the Rosenbrock method (4th order of accuracy), Radau, BDF, and LSODA. The Rozobrok method is implemented in the Python programming language, the rest of the methods were taken from the Scipy Python library. The paper describes each numerical method, which makes it possible to justify the choice of methods for solving the problem. The main purpose of the work is a comparative analysis of their effectiveness according to the criteria of accuracy, stability and computational complexity. The order of accuracy of the methods using the Runge rule is estimated in Python, and their characteristics are analyzed when solving a system of two linear ordinary differential equations of the second order with non-constant coefficients. The paper presents graphs of the dependence of the order of accuracy on the step of calculations, waveforms of solutions and phase trajectories of a mathematical model that clearly demonstrate the behavior of the system under various parameters. Special attention is paid to the adaptability of the methods to the rigidity of the system due to the presence of rapidly attenuating and high-frequency components. The results show that the Rosenbrock method provides high accuracy with an analytically specified Jacobi matrix, while the order of the other methods has an experimental order lower than the theoretical one. The data obtained allows us to determine the optimal modeling method depending on the required accuracy and computing resources. The study contributes to the development of numerical approaches to the analysis of geoacoustic processes and can be used in predicting deformation phenomena in rocks.