Finite Difference Models of Dynamical Systems with Polynomial Right-Hand Side

This paper presents a detailed examination of difference schemes that establish a one-to-one correspondence between time layers, commonly known as Kahan’s method or reversible difference schemes. These mathematical frameworks play a crucial role in numerical analysis and computational mathematics, enabling accurate modeling of dynamic systems. The paper discusses the applications of these schemes, particularly in dynamic systems with quadratic right-hand sides, which are found in various fields such as physics, engineering, and applied mathematics. These systems often describe complex phenomena, including mechanical vibrations and fluid dynamics. Additionally, the study explores the integration of Kahan’s method with the direct method for solving partial differential equations (PDEs) in mathematical physics. This combination aims to enhance the accuracy and computational efficiency of numerical solutions. By investigating the correlation between these methodologies, this work seeks to advance numerical techniques for addressing complex dynamic systems. The findings indicate that this integration improves the stability and convergence of solutions, highlighting the potential of Kahan’s method and reversible difference schemes in tackling challenges across diverse scientific disciplines.