Energy-stepping integrators in Lagrangian mechanics
Marcial Gonzalez, Bernd Schmidt and Michael Ortiz
Int. J. Numerical Methods in Engng., Vol. 82, 205-241, 2010
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Abstract
We present a class of integration schemes for Lagrangian mechanics, referred to as
energy-stepping integrators, that are momentum and energy conserving, symplectic
and convergent. In order to achieve these properties we replace the original potential
energy by a piecewise constant, or terraced approximation at steps of uniform
height. By taking steps of diminishing height, an approximating sequence of energies is
generated. The trajectories of the resulting approximating Lagrangians can be characterized
explicitly and consist of intervals of piecewise rectilinear motion. We show that the
energy-stepping trajectories are symplectic, exactly conserve all the momentum maps of the
original system and, subject to a transversality condition, converge to trajectories of the
original system when the energy step is decreased to zero. These properties, the excellent
long-term behavior of energy-stepping and its automatic time-step selection property are
born out by selected examples of application, including the dynamics of a frozen Argon
cluster, the spinning of an elastic cube and the collision of two elastic spheres.
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