On non-Hermitian positive (semi)definite linear algebraic systems arising from dissipative Hamiltonian DAEs

TL;DR

This paper analyzes linear algebraic systems from dissipative Hamiltonian DAEs, focusing on non-Hermitian matrices with positive (semi)definite Hermitian parts, and discusses iterative solution methods and their performance.

Contribution

It characterizes the structure of linear systems from dissipative Hamiltonian DAEs and demonstrates effective iterative methods for solving them, including handling singular parts.

Findings

Iterative Krylov methods are effective for positive definite cases.

Handling semidefinite cases requires additional techniques.

Performance illustrated on practical examples.

Abstract

We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An important feature of the linear algebraic systems is that the (non-Hermitian) system matrix has a positive definite or semidefinite Hermitian part. In the positive definite case we can solve the linear algebraic systems iteratively by Krylov subspace methods based on efficient three-term recurrences. We illustrate the performance of these iterative methods on several examples. The semidefinite case can be challenging and requires additional techniques to deal with "singular part", while the "positive definite part" can still be treated with the three-term recurrence methods.