Variational symplectic diagonally implicit Runge-Kutta methods for isospectral systems

TL;DR

This paper develops variational symplectic diagonally implicit Runge-Kutta methods for isospectral systems, ensuring eigenvalue conservation and energy preservation, with practical implementation advantages demonstrated through numerical experiments.

Contribution

It introduces a Lagrangian formulation of isospectral Runge-Kutta methods via discrete Euler-Poincare reduction, generalizing previous Lie group integrators and enabling higher order methods.

Findings

Eigenvalues are conserved with high accuracy.

Energy conservation is achieved in numerical experiments.

Methods are straightforward to implement for higher order accuracy.

Abstract

Isospectral flows appear in a variety of applications, e.g. the Toda lattice in solid state physics or in discrete models for two-dimensional hydrodynamics, with the isospectral property often corresponding to mathematically or physically important conservation laws. Their most prominent feature, i.e. the conservation of the eigenvalues of the matrix state variable, should therefore be retained when discretizing these systems. Recently, it was shown how isospectral Runge-Kutta methods can, in the Lie-Poisson case also considered in our work, be obtained through Hamiltonian reduction of symplectic Runge-Kutta methods on the cotangent bundle of a Lie group. We provide the Lagrangian analogue and, in the case of symplectic diagonal implicit Runge-Kutta methods, derive the methods through a discrete Euler-Poincare reduction. Our derivation relies on a formulation of diagonally implicit isospectral Runge-Kutta methods in terms of the Cayley transform, generalizing earlier work that showed this for the implicit midpoint rule. Our work is also a generalization of earlier variational Lie group integrators that, interestingly, appear when these are interpreted as update equations for intermediate time points. From a practical point of view, our results allow for a simple implementation of higher order isospectral methods and we demonstrate this with numerical experiments where both the isospectral property and energy are conserved to high accuracy.