On a set of tests for numerical methods of integrating differential equations, based on the Calogero system

Abstract

Based on the completely integrable Calogero dynamical system, which describes the one-dimensional many-body problem, a tool for testing difference schemes has been developed and implemented in the original fdm package integrated into the Sage computer algebra system. This work shows how the developed tools can be used to examine the behavior of numerical solutions near the collision point and how to study the conservatism of the difference scheme. When detecting singularities using Alshina’s method, a difficulty was discovered associated with false order fluctuations. One of the main advantages of this set of tests is the purely algebraic nature of the solutions and integrals of motion.