On different methods to construct reversible difference schemes

This paper explores di erent approaches to constructing di erence schemes for solving di erential equations, including rst-order nonlinear hyperbolic systems and other dynamical systems, such as the Jacobi oscillator and the Weierstrass oscillator. We also consider various initial conditions and their impact on the numerical solutions. One approach we examine is the multi-component method of alternating directions, which has been widely used in multidimensional mathematical physics. This method is particularly useful because it is unconditionally stable, computationally e cient, and does not require stabilizing corrections. However, while it works well for many problems, it is not the only method available. Another approach for nding the system's solution is the combination of Appelroth's quadratization technique with Kahan's discretization method. This, in turn, results in a di erence scheme that establishes a one-to-one correspondence (Cremona map) between the initial and nal states, improving the precision of the solution. To provide a broader comparison, it is also worth considering the widely used Runge-Kutta method, known for its high accuracy in solving di erential equations. By comparing Runge-Kutta schemes with Kahan's method and the alternating direction approach, we highlight key di erences in their computational e ciency, stability, and accuracy. The aim of this work is to provide a clearer understanding of how di erent numerical methods perform and outline when each approach may be most e ective.