Difference schemes with point symmetries and their numerical tests

TL;DR

This paper introduces symmetry-preserving difference schemes for second and third order ODEs that maintain the original equations' symmetries, resulting in more accurate numerical solutions, especially near singularities.

Contribution

The paper develops new difference schemes that preserve the symmetry groups of differential equations, enhancing accuracy and robustness over standard methods.

Findings

Symmetry-preserving schemes outperform standard methods in accuracy.

They provide valid solutions beyond singular points.

Numerical tests confirm improved performance.

Abstract

Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new difference schemes are tested as numerical methods. The obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost. For an example involving a solution with a singularity in the integration region the symmetry preserving scheme, contrary to standard ones, provides solutions valid beyond the singular point.