Variational symplectic diagonally implicit Runge-Kutta methods for isospectral systems

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.

READ FULL TEXT

page 1

page 2

page 3

page 10

page 11

page 12

page 14

research
11/20/2022

Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery

By one of the most fundamental principles in physics, a dynamical system...
research
06/24/2019

Variational order for forced Lagrangian systems II: Euler-Poincaré equations with forcing

In this paper we provide a variational derivation of the Euler-Poincaré ...
research
04/21/2023

On discrete analogues of potential vorticity for variational shallow water systems

We outline how discrete analogues of the conservation of potential vorti...
research
01/19/2021

Multisymplectic Hamiltonian Variational Integrators

Variational integrators have traditionally been constructed from the per...
research
05/17/2022

Large-stepsize integrators with improved uniform accuracy and long time conservation for highly oscillatory systems with large initial data

In this paper, we are concerned with large-stepsize highly accurate inte...
research
07/17/2018

From modelling of systems with constraints to generalized geometry and back to numerics

In this note we describe how some objects from generalized geometry appe...

Please sign up or login with your details

Forgot password? Click here to reset