Exponential Variational Integrators Using Constant or Adaptive Time Step

In this book article, at first we survey some recent advances in variational integrators focusing on the class of them known as exponential variational integrators, applicable in finite dimensional mechanical systems. Since these integrators are based on the space and time discretization, we start with a brief summary of the general development of the discrete mechanics and its application in describing mechanical systems with space-time integration algorithms. We, then, make an attempt to treat briefly in depth only the particular topic of adaptive time step exponential variational integrators. To this aim, the action integral along any curve segment is defined using a discrete Lagrangian that depends on the endpoints of the segment and on a number of intermediate points of interpolation. This Lagrangian is then, at any time interval, written as a weighted sum of the Lagrangians corresponding to a set of the chosen intermediate points to obtain high order integrators. The positions and velocities are interpolated here using special exponential functions. Finally, we derive exponential higher order variational integration methods for the numerical integration of systems with oscillatory solutions. The obtained exponential variational integrators using constant or adaptive time step are tested for the numerical solution of several problems showing their good behavior to track oscillatory solutions. Furthermore, we use the space-time geodesic approach of classical mechanics to explore whether the new methodology may be effective in adaptive time schemes.