Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems

Adrian Viorel; Cristian D. Alecsa; Titus O. Pinţa

doi:10.3934/dcds.2020407

Article Contents

2021, Volume 41, Issue 7: 3319-3341. Doi: 10.3934/dcds.2020407

This issue Previous Article Complex planar Hamiltonian systems: Linearization and dynamics Next Article Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains

Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems

1.
Babeş-Bolyai University Cluj-Napoca, Str. Kogalniceanu 3, 400084 Cluj-Napoca, Romania
2.
Romanian Institute of Science and Technology, 400022 Cluj-Napoca, Romania and, Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, 400320 Cluj-Napoca, Romania
3.
Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany

^* Corresponding author: A. Viorel
^* Corresponding author: A. Viorel

Received: April 30, 2020

Revised: August 31, 2020

Early access: December 2020

Published: July 2021

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Abstract

The present work deals with the numerical long-time integration of damped Hamiltonian systems. The method that we analyze combines a specific Strang splitting, that separates linear dissipative effects from conservative ones, with an energy-preserving averaged vector field (AVF) integrator for the Hamiltonian subproblem. This construction faithfully reproduces the energy-dissipation structure of the continuous model, its equilibrium points and its natural Lyapunov function. As a consequence of these structural similarities, both the convergence to equilibrium and, more interestingly, the energy decay rate of the continuous dynamical system are recovered at a discrete level. The possibility of replacing the implicit AVF integrator by an explicit Störmer-Verlet one is also discussed, while numerical experiments illustrate and support the theoretical findings.

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Mathematics Subject Classification: Primary: 37M15, 65P10; Secondary: 34D05.

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Figure 1. The (eventual) exponential decay of the potential energy (a) as well as state-spaces trajectories (b) for decreasing step sizes $ h = 1 \text{ and } 0.1 $, both with identical initial conditions $ u_0 = 0.01, v_0 = 0 $. The double-well potential is depicted in c)

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Figure 2. Capturing an energy plateau: a) the AVF Splitting algorithm (black) compared to the trapezoidal rule (blue) for $ h = 1 $ (and benchmark Runge-Kutta (red)); b) AVF Splitting algorithm (black) compared to the conformal symplectic algorithm (28) (green) for $ h = 0.1 $. The contour lines of the nonconvex potential (33) are depicted in c)

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Figure 3. Erroneous total energy oscillations: a) AVF Splitting algorithm (black) compared to the trapezoidal rule (blue) for $ h = 0.1 $ (and benchmark Runge-Kutta (red)); b) AVF Splitting algorithm (black) compared to the conformal symplectic algorithm (28) (green) for $ h = 0.01 $. The contour lines of the Rosenbrock potential are depicted in c)

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