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

  • * Corresponding author: A. Viorel

    * Corresponding author: A. Viorel 
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  • 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.

    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)

    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)

    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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