Low-Rank Solvers for Energy-Conserving Hamiltonian Boundary Value Methods
For researchers and practitioners solving Hamiltonian systems that require long-term energy conservation, this work offers a more efficient computational approach, though it is an incremental improvement on existing HBVMs.
This paper introduces low-rank solvers for Hamiltonian Boundary Value Methods (HBVMs), exploiting the low-rank structure of the stage equations to accelerate both linear and nonlinear solves. Numerical experiments on wave equations show improved efficiency and robustness.
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.