Kaliukh I., Trofymchuk O., Lebid O.
Peculiarities of applying the finite-difference method for calculation of nonlinear problems of the dynamics of distributed systems in a flow

The peculiarities of applying the finite-difference method (FDM) for solving nonlinear dynamic problems of distributed systems (DS) in a flow are considered. The main limitations for using the FDM for numerical modeling of wave propagation and reflection in DS are shown to be the special features of the constitutive quasilinear equations. They necessitate the simultaneous calculation of the variables corresponding to fast and slow wave processes. The term “singularly perturbed system of equations” is used for such systems. These perturbations are the result of a significant difference in the propagation velocities of longitudinal, configurational, bending, and torsional waves in the DS at the physical level. Therefore, it is necessary to use special step-by-step methods of regularization and filtering of the numerical results. It imposes certain contstraints on the ability to model real processes and the accuracy of the results and forces the use of implicit difference schemes and high-frequency filtering. When solving the system of linear algebraic equations, taking into account the poor conditioning of the matrix of convective terms, the method of regularization was chosen by experimental calculation. Calculations according to the Crank–Nicolson difference scheme, even using coarse grids, can give the results with the required degree of accuracy and minimal time cost. Another picture is observed when comparing the results on coarse and fine grids for the Euler difference scheme. Inherent errors brought in by errors in approximating the missing boundary conditions lead to greater differences. © 2023, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords: distributed systems, finite-difference method, high-frequency filtering, nonlinearity, singularity, Control nonlinearities, Linear algebra, Linear equations, Numerical methods, Wave propagation, Coarse grid, Difference schemes, Distributed systems, Finite-difference methods, Frequency filtering, High frequency HF, High-frequency filtering, Method of regularization, Nonlinearity, Singularity, Finite difference method

Cite:
Kaliukh I., Trofymchuk O., Lebid O. (2023). Peculiarities of applying the finite-difference method for calculation of nonlinear problems of the dynamics of distributed systems in a flow. Cybernetics and Systems Analysis, 59 (1), 139–155. doi: https://doi.org/10.1007/s10559-023-00548-4 http://jnas.nbuv.gov.ua/article/UJRN-0001380511 [In Ukrainian].