Numerical scheme for one integro-differential system related to the problem of space dynamo

E. A. Kazakov

Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 49 (2024) no. 4, pp. 85-98

Résumé

The article is devoted to the description of the developed numerical scheme for modeling a hereditary dynamic system, which is a model of a two-mode hydromagnetic dynamo. The models include two magnetic field generators – large-scale and turbulent ($\alpha$-effect). The influence of the magnetic field on the motion of the medium is presented through the suppression of the $\alpha$-effect by the functional of the field components, which introduces memory into the model (hereditary). The model is described by an integro-differential system of equations. The paper presents the numerical scheme itself and investigates the order of accuracy on nested grids. The numerical scheme consists of two parts, the trapezoid method is used for the differential part, and the trapezoid quadrature formula is used for the integral part. As a result of conjugation of the schemes, we obtain a nonlinear algebraic system of equations. To solve such a system, it is necessary to involve methods for nonlinear algebraic systems. In this paper, the Newton method was chosen. It is shown that in the case of an exponential kernel of the suppression functional, the model can be reduced to the classical Lorenz system. The known nature of the dynamics of the Lorenz system for various parameters allowed us to verify the numerical scheme. It is shown that the numerical scheme allows us to qualitatively solve the integro-differential system of equations, which is a model of a cosmic dynamo. This numerical scheme was developed for a specific model, but can be easily generalized for other quadratic-nonlinear integro-differential systems.

Détail

Keywords: hydromagnetic dynamo, systems with memory, heredity, integro-differential equations, numerical scheme
Mots-clés : Volterra vector equation.