Why geometric integration?
This is a reflective position paper for researchers in numerical analysis and GNI, evaluating the field's progress and future direction.
The paper assesses the current state of geometric numerical integration (GNI), questioning whether its major challenges have been solved and whether it should remain a separate field or be absorbed into broader numerical analysis.
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?