Systematic Discovery of Runge-Kutta Methods through Algebraic Varieties

TL;DR

This paper introduces an evolutionary optimization algorithm to systematically discover Runge-Kutta methods by modeling their algebraic properties as algebraic varieties, enabling efficient design and optimization.

Contribution

It presents a novel approach linking algebraic geometry with evolutionary algorithms to optimize Runge-Kutta methods of arbitrary order.

Findings

Successful design of explicit s-level Runge-Kutta methods of order q

Demonstrated the effectiveness of the algebraic variety approach

Open potential for modeling other algebraic problems in numerical analysis

Abstract

This work presents a new evolutionary optimization algorithm in theoretical mathematics with important applications in scientific computing. The use of the evolutionary algorithm is justified by the difficulty of the study of the parametrization of an algebraic variety, an important problem in algebraic geometry. We illustrate an application, Evo-Runge-Kutta, in a problem of numerical analysis. Results show the design and the optimization of particular algebraic variety, the explicit s levels Runge-Kutta methods of order q. The mapping between algebraic geometry and evolutionary optimization is direct, and we expect that many open problems will be modelled in the same way.