Quadratic projectable Runge-Kutta methods

Runge-Kutta methods are affine equivariant: applying a method before or after an affine change of variables yields the same numerical trajectory. However, for some applications, one would like to perform numerical integration after a quadratic change of variables. For example, in Lie-Poisson reduction, a quadratic transformation reduces the number of variables in a Hamiltonian system, yielding a more efficient representation of the dynamics. Unfortunately, directly applying a symplectic Runge-Kutta method to the reduced system generally does not preserve its Hamiltonian structure, so many proposed techniques require computing numerical trajectories of the original, unreduced system. In this paper, we study when a Runge-Kutta method in the original variables descends to a numerical integrator expressible entirely in terms of the quadratically transformed variables. In particular, we show that symplectic diagonally implicit Runge-Kutta (SyDIRK) methods, applied to a quadratic projectable vector field, are precisely the Runge-Kutta methods that descend to a method (generally not of Runge-Kutta type) in the projected variables. We illustrate our results with several examples in both conservative and non-conservative dynamics.

💡 Research Summary

The paper investigates the problem of performing numerical integration after a quadratic change of variables, a situation that frequently arises in applications such as Lie‑Poisson reduction where a quadratic momentum map reduces the dimensionality of a Hamiltonian system. While symplectic Runge–Kutta (RK) methods are known to be affine‑equivariant, they generally do not preserve the Hamiltonian structure when applied directly to the reduced (projected) system. Consequently, many existing techniques first integrate the original high‑dimensional system and then project the solution, which incurs substantial computational cost.

The authors formalize the notion of an F‑projectable vector field: a vector field (f) on a space (Y) is F‑projectable if there exists a vector field (g) on the target space (Z) such that (F’(y)f(y)=g(F(y))) for a given quadratic map (F:Y\to Z). Under this relation, continuous trajectories satisfy (z(t)=F(y(t))) and the reduced dynamics obey (\dot z=g(z)). The central question is whether a Runge–Kutta method applied to (\dot y=f(y)) can be “descended” to a method that uses only the projected variables (z), without ever computing the intermediate stages (Y_i) in the original space.

Starting from the standard s‑stage RK scheme, \