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Modified equations for variational integrators applied to Lagrangians linear in velocities

This research was supported by the DFG Collaborative Research Center TRR 109, "Discretization in Geometry and Dynamics"

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  • Variational integrators applied to degenerate Lagrangians that are linear in the velocities are two-step methods. The system of modified equations for a two-step method consists of the principal modified equation and one additional equation describing parasitic oscillations. We observe that a Lagrangian for the principal modified equation can be constructed using the same technique as in the case of non-degenerate Lagrangians. Furthermore, we construct the full system of modified equations by doubling the dimension of the discrete system in such a way that the principal modified equation of the extended system coincides with the full system of modified equations of the original system. We show that the extended discrete system is Lagrangian, which leads to a construction of a Lagrangian for the full system of modified equations.

    Mathematics Subject Classification: 37M15, 65L06, 70H03.


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  • Figure 1.  Pendulum with midpoint rule (left) and trapezoidal rule (right), both with step size $ h = 0.35 $ and initial point $ (3,0) $ (top) and $ (1.5,0) $ (bottom).
    Dashed curve: exact solution.
    Bullets: discrete solution.
    Solid curve: solution of the principal modified equation, truncated after second order.
    Line segments: visualization of parasitic oscillations

    Figure 2.  Leapfrogging vortex pairs with the midpoint rule. No parasitic behavior is visible

    Figure 3.  Leapfrogging vortex pairs with the trapezoidal rule. One observes parasitic oscillations

    Figure 4.  Enlarged versions of the right hand sections of Figures 2-3: midpoint rule (left) and trapezoidal rule (right)

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