Why geometric integration?

TL;DR

Geometric numerical integration (GNI) has become a dominant approach in solving differential equations, prompting reflection on its achievements, current relevance, and future directions in the field.

Contribution

The paper critically examines the evolution, impact, and ongoing significance of GNI in numerical analysis, questioning its future role and potential for further development.

Findings

GNI has established itself as a major paradigm in numerical ODEs and PDEs.

The field has achieved significant milestones, but questions remain about its future trajectory.

GNI's influence has enabled new research directions and interdisciplinary applications.

Abstract

Since its emergence, GNI has become the new paradigm in numerical solution of ODEs, while making significant inroads into numerical PDEs. As often, yesterday's revolutionaries became the new establishment. This is an excellent moment to pause and take stock. Have all the major challenges been achieved, all peaks scaled, leaving just a tidying-up operation? Is there still any point to GNI as a separate activity or should it be considered as a victim of its own success and its practitioners depart to fields anew - including new areas of activity that have been fostered or enabled by GNI?