2026 , Volume 31, № 3, p.24-39

The study examines the convergence of solution methods applied to fully conservative difference schemes in three-dimensional magnetohydrodynamics (MHD).The methods are focused on combined and separate approaches to solving systems of difference equations separated by physical processes.

During the research, convergence estimates of iterative processes for three-dimensional MHD difference equations were established and analyzed. These theoretical estimates were rigorously derived and confirmed through numerical calculations. Precise conditions for the effective application of these methods were determined. A series of computational experiments was conducted to verify the obtained results. A comprehensive approach to analyzing the stability of numerical solutions was developed.

The research led to several key results. Detailed convergence estimates of iterative processes were obtained and mathematically confirmed. The applicability domain of combined and separate solution methods was determined. Numerical calculations confirmed the validity of theoretical estimates. Analysis of the influence of various parameters on convergence rate and solution stability was particularly examined.

The results have both theoretical and practical significance.

The developed methodology allows selecting both optimal solution methods at each time step and significantly reducing computation time. Practical implementation demonstrated a significant increase in computational efficiency when conducting large-scale numerical experiments. The obtained results can be used to optimize computational processes in various areas of MHD modelling. The developed methods have been successfully applied to various applied problems in magnetic hydrodynamics, including modelling of plasma processes and magnetohydrodynamic flows in technological installations. The results have been implemented in software packages for numerical simulation of MHD processes and have shown high efficiency in practice. The created algorithms allow significant accelerating the process of numerical modelling of complex physical systems.