Symplectic Isospectral Runge--Kutta Methods as Lie group methods
This work provides an incremental improvement in numerical methods for Hamiltonian dynamics, benefiting researchers in computational mathematics and physics.
The paper tackled the problem of structure-preserving numerical integration for isospectral flows on quadratic Lie algebras, showing that applying symplectic Runge-Kutta methods to a transport formulation on the Lie group is equivalent to existing ISOSYRK schemes and is more efficient for multi-stage methods.
We compare three approaches for structure preserving numerical integration of isospectral flows on quadratic Lie algebras. Such flows originate from Hamiltonian dynamics on the cotangent bundle of the Lie group. It is known, via discrete reduction theory, that symplectic Runge--Kutta methods applied to the cotangent bundle formulation induce isospectral symplectic Runge--Kutta (ISOSYRK) schemes on the Lie algebra. Here, we show that the same symplectic Runge--Kutta method, but applied to the transport formulation of the flow on the Lie group, is equivalent to the corresponding ISOSYRK scheme. We also give numerical results suggesting that the formulation on the Lie group is more efficient for schemes with two or more intermediate stages.