Nonlinear wave model of towed system and numerical method for its calculation
DOI:
https://doi.org/10.32347/2411-4049.2024.3.102-111Keywords:
mathematical modeling, towed systems, waves, finite difference method, stability of computationAbstract
Towed systems have found wide practical applications. Notable underwater systems for the long-distance extraction of minerals (concretions) from the ocean floor extend over 5-10 km. The existing mathematical models for solving dynamic problems of such systems in various environments are not entirely accurate regarding the diversity of wave processes. This necessitates the development of refined wave models. This article presents a new quasi-linear mathematical model describing the nonlinear four-mode dynamics of a towed system in a spatially inhomogeneous field of mass and surface forces. It is described by a nonlinear system of twelve first-order partial differential equations. The principles of boundedness and hyperbolicity are satisfied. The validation of the two-mode reduction of the model is based on the numerical solution of the problem of the propagation of two waves: longitudinal and configurational. Using a numerical algorithm and a program based on the finite difference method, a comparison of two difference schemes – Crank-Nicolson and Euler – was conducted. The main limitations for applying the finite difference method used for numerical modeling of wave propagation and reflection in a towed system are the peculiarities of the defining quasi-linear equations, which are related to the necessity of simultaneous computation of variables responsible for fast and slow processes. For such systems of equations, the term "singularly perturbed system of equations" is used. These perturbations result from the significant differences in the propagation speeds of longitudinal and configurational waves at the physical level. Consequently, it is necessary to employ special time-stepping regularization and filtering methods for numerical results. This imposes certain restrictions on the ability to model real processes and on the accuracy of the obtained results, necessitating the use of implicit difference schemes and high-frequency filtering.
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