### Variational symplectic diagonally implicit Runge-Kutta methods for isospectral systems

Clauson Carvalho da Silva and Christian Lessig

BIT Numerical Mathematics, 2022

Comparison between the iospectral variational time integrator in our work and the structure preserving time integrator from (E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden, and M. Desbrun. Geometric, variational discretization of continuum theories. Physica D: Nonlinear Phenomena, 240(21), 2011). On the top is energy (which is conserved in the continuous system) and at the bottom the eigenvalues of the angular momentum matrix (whose conservation amounts to the isospectral property).

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.

##### Resources

Preprint

Reference implementation Rigid body (Mathematica)

Reference implementation Toda Lattive (Mathematica)

Reference implementation Barotropic Vorticity Equation (Mathematica)

@article{Silva2022,

author = {da Silva, Clauson Carvalho and Lessig, Christian},

doi = {10.1007/s10543-022-00932-0},

isbn = {1572-9125},

journal = {BIT Numerical Mathematics},

title = {Variational symplectic diagonally implicit Runge-Kutta methods for isospectral systems},

url = {https://doi.org/10.1007/s10543-022-00932-0},

year = {2022}}

##### Acknowledgement

We would like to thank Mathieu Desbrun for helpful discussions.