Dirichlet-to-Neumann boundary condition for time-dependent dispersive waves in three-dimensional guides (Q1880743)

The dispersive Klein-Gordon equation is considered in a three-dimensional semi-infinite cylindrical domain (in a waveguide). The dependence between the boundary values of solutions and their normal derivatives on the cross-section of a waveguide is derived, this dependence is called as Dirichlet-to-Neumann (DtN) boundary condition. The DtN boundary condition is nonlocal in space but is local in time. An arbitrary unbounded part of the waveguide can be truncated and replaced by the DtN condition on the auxiliary artifical section. Thus, the boundary value problem in semi-infinite domain is reduced to the problem in the bounded domain which contains all irregularities of the waveguide. The truncated condition is incorporated in a finite element scheme to solve the problem in the finite domain. The complete set of semi-discrete equations is described. Some numerical examples are given and discussed.