Low-Rank Solvers for Energy-Conserving Hamiltonian Boundary Value Methods

TL;DR

This paper introduces low-rank solvers for energy-conserving Hamiltonian Boundary Value Methods, enhancing efficiency and robustness in long-term simulations of Hamiltonian systems.

Contribution

It develops low-rank Krylov-based algorithms tailored for HBVMs, improving computational efficiency for both linear and nonlinear Hamiltonian problems.

Findings

Efficient Krylov solvers for linear HBVMs.

Effective Newton-Krylov methods for nonlinear HBVMs.

Numerical experiments show robustness and efficiency.

Abstract

We study energy-conserving Hamiltonian Boundary Value Methods (HBVMs) for Hamiltonian systems, which arise in applications where long-term preservation of energy and symplecticity is essential. HBVMs are multi-stage schemes whose stage equations reformulate as matrix equations with a low-rank right-hand side. For linear systems, we exploit this structure directly via Krylov projection solvers. For nonlinear systems, we leverage it within simplified Newton iterations and as a preconditioner in a Newton--Krylov framework, combined with adaptive time-stepping for robust convergence. Numerical experiments on semi-discretized wave equations demonstrate the efficiency and robustness of the proposed approach.