Olivier Verdier: What is unique about Runge–Kutta methods?

Runge–Kutta methods are numerical integrators for ordinary differential equations. RK methods (and Butcher series, their modern counterpart) have been used by mathematicians and engineers for over a hundred years. They are used in such diverse applications as quantum mechanics (renormalisation), control theory, Hamiltonian Monte Carlo methods, or averaging. Yet, there is no understanding of what is really unique about RK methods and Butcher series. Due to recent exciting new developments, an answer to that question is now available: RK methods are, in a precise sense which I will explain, the only integrators compatible with affine transformations. This in turn sheds new light on numerical integrators in general, and on the new role played by symmetry groups in numerical analysis. It also leads to a new class of integrators: the aromatic Runge–Kutta methods, which can overcome some of the shortcomings of RK methods. The talk will be kept at an elementary level, and no prerequisite knowledge is necessary to follow the talk.