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

TL;DR

Energy-conserving Hamiltonian Boundary Value Methods (HBVMs) address long-term preservation of energy and symplectic structure in Hamiltonian systems. The paper develops low-rank solution strategies: Krylov projection for linear HBVM stage equations and a Newton–Krylov framework with a low-rank matrix-equation preconditioner for nonlinear problems, aided by adaptive time stepping. Numerical experiments on semi-discretized semilinear and linear wave equations show robust convergence with few outer iterations, even for higher-order HBVM configurations. The work suggests extensions to Caputo fractional derivatives, distributed implementations, and applications to other structure-preserving integrators, with considerations for optimized pole selection to accelerate convergence.

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.

Figures (2)

• Figure 1.4: Nonlinear test case 1 ($s=2, k=3$). The top figure reports an histogram of the number of Newton iterations per time-step on the left $y$-axis, while on the right there is the residual measured as the norm of $\|\mathcal{F}(\Phi)\|_F$. On the left plot of the second row we have an heatmap with the number of FGMRES iterations preconditioned with the matrix equation solver preconditioner per time step VS Newton iteration; on the right there is the corresponding FGMRES residual, showing the adaptive selection of the tolerance along the Newton iteration.
• Figure 1.5: Nonlinear test case 2 ($s=3, k=6$). The top figure reports an histogram of the number of Newton iterations per time-step (averaged every 5 time-steps) on the left $y$-axis, while on the right there is the residual measured as the norm of $\|\mathcal{F}(\Phi)\|_F$ for each time-step. On the left plot of the second row we have an heatmap with the number of FGMRES iterations preconditioned with the matrix equation solver preconditioner per time-step VS Newton iteration; on the right there is the corresponding FGMRES residual, showing the adaptive selection of the tolerance along the Newton iteration. In the bottom panels, we have cropped the number of Newton iterations to 40, consistently with the top panel.