A flexible short recurrence Krylov subspace method for matrices arising in the time integration of port Hamiltonian systems and ODEs/DAEs with a dissipative Hamiltonian

TL;DR

This paper introduces a new class of short recurrence Krylov subspace methods that efficiently solve matrices from port Hamiltonian systems and ODEs/DAEs by allowing approximate solutions of the Hermitian part within each iteration.

Contribution

It develops right preconditioned variants of Krylov methods that enable approximate solutions of the Hermitian part, enhancing efficiency for large-scale dissipative Hamiltonian systems.

Findings

Methods are effective for large-scale systems.

Allows use of few multigrid or CG steps per iteration.

Numerical experiments demonstrate improved efficiency.

Abstract

For several classes of mathematical models that yield linear systems, the splitting of the matrix into its Hermitian and skew Hermitian parts is naturally related to properties of the underlying model. This is particularly so for discretizations of dissipative Hamiltonian ODEs, DAEs and port Hamiltonian systems where, in addition, the Hermitian part is positive definite or semi-definite. It is then possible to develop short recurrence optimal Krylov subspace methods in which the Hermitian part is used as a preconditioner. In this paper we develop new, right preconditioned variants of this approach which as their crucial new feature allow the systems with the Hermitian part to be solved only approximately in each iteration while keeping the short recurrences. This new class of methods is particularly efficient as it allows, for example, to use few steps of a multigrid solver or a (preconditioned) CG method for the Hermitian part in each iteration. We illustrate this with several numerical experiments for large scale systems.