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

Abstract

Exact Dirichlet-to-Neumann (DtN) boundary conditions on cross-sections of three-dimensional semi-infinite wave guides are derived. This enables the exact truncation of a wave guide to allow computations in a finite domain Ω. The DtN boundary condition is nonlocal in space but local in time. Practical implementation requires the truncation of the exact boundary condition by approximating an infinite sum with a finite sum, and by terminating an infinite recursion relation. The truncated condition is incorporated in a finite element scheme to solve the problem in Ω. The cross-section of the guide may have an arbitrary shape. The three-dimensional time-dependent dispersive wave equation is considered in the guide. The dispersion parameter is allowed to vary in the cross-section. All the results reduce immediately to the two-dimensional and to the one-dimensional cases. To the best of the authors' knowledge, this is the first time that an exact boundary condition is derived in the dispersive case, even in one dimension.