Hopf algebra techniques to handle dynamical systems and numerical integrators
This work provides a theoretical extension for researchers in numerical analysis and dynamical systems, but it is incremental as it generalizes existing algebraic frameworks without demonstrating new practical breakthroughs.
The authors extend algebraic techniques based on Hopf algebras to analyze dynamical systems and numerical integrators, enabling more efficient computations by generalizing previous methods that used specific Hopf algebras.
In a series of papers the present authors and their coworkers have developed a family of algebraic techniques to solve a number of problems in the theory of discrete or continuous dynamical systems and to analyze numerical integrators. Given a specific problem, those techniques construct an abstract, {\em universal} version of it which is solved algebraically; then, the results are tranferred to the original problem with the help of a suitable morphism. In earlier contributions, the abstract problem is formulated either in the dual of the shuffle Hopf algebra or in the dual of the Connes-Kreimer Hopf algebra. In the present contribution we extend these techniques to more general Hopf algebras, which in some cases lead to more efficient computations.