Discrete Transparent Boundary Conditions for the Equation of Rod Transverse Vibrations

Abstract

Local perturbations of an infinitely long rod go away to infinity. On the contrary, in the case of a finite length of the rod, the perturbations reach its boundary and are reflected from it. The boundary conditions constructed here for the implicit difference scheme imitate the Cauchy problem and provide almost no reflection. These boundary conditions are non-local in time, and their practical implementation requires additional calculations at every time step. To minimise them, a special rational approximation similar to the Hermite - Pad\'e approximation is used. Numerical experiments confirm the high "transparency" of these boundary conditions and determine the conditional stability regions for finite-difference scheme.