  We present a mathematical and numerical investigation to the shrinkingdimer
saddle dynamics for finding any-index saddle points in the solution landscape.
Due to the dimer approximation of Hessian in saddle dynamics, the local
Lipschitz assumptions and the strong nonlinearity for the saddle dynamics, it
remains challenges for delicate analysis, such as the the boundedness of the
solutions and the dimer error. We address these issues to bound the solutions
under proper relaxation parameters, based on which we prove the error estimates
for numerical discretization to the shrinking-dimer saddle dynamics by matching
the dimer length and the time step size. Furthermore, the Richardson
extrapolation is employed to obtain a high-order approximation.
  The inherent reason of requiring the matching of the dimer length and the
time step size lies in that the former serves a different mesh size from the
later, and thus the proposed numerical method is close to a fully-discrete
numerical scheme of some spacetime PDE model with the Hessian in the saddle
dynamics and its dimer approximation serving as a "spatial operator" and its
discretization, respectively, which in turn indicates the PDE nature of the
saddle dynamics.
