Local High-Order Absorbing Boundary Conditions for Time-Dependent Waves in Guides

Abstract

The scalar wave equation in a two-dimensional semi-infinite wave guide is considered. The recently proposed Hagstrom–Warburton (H–W) local high-order absorbing boundary conditions (ABCs), which are based on a modification of the Higdon ABCs, are presented in this context. The P-order ABC involves the free parameters 0 < aj ≤ 1, for j = 0, 1, …, P, which have to be chosen. The choice aj = 1 for all j is shown to be satisfactory, in general, although not necessarily optimal. The optimal choice of the parameters is discussed via both theoretical analysis and numerical experiments. In addition, an adaptive scheme which controls the time-varying values of P and aj is presented and tested.

Equivalent Boundary Conditions for Heterogeneous Acoustic Media

“…The boundary condition (5) is a Higdon BC [9] and coefficients a j ∈ (0, 1] are given parameters. Rewriting (5) according to the Hagstrom-Warburton formulation [10,14,15], adapted to the problem stated in the frequency domain, BC (5) becomes…”

“…Among many possibilities, one could use a non-reflecting BC, defined through pseudo-differential operators [1,2,3,4], or choose from a variety of approximate BCs, such as PMLs [5,6,7] or Absorbing Boundary Conditions (ABCs) [8,9,10,11,12,13]. In this work the Hagstrom-Warburton ABC [10,14] is chosen. The benefit of implementing nonreflecting BCs is discussed in [8] and it has been proven in [15] that the Hagstrom-Warburton ABC is equivalent to the popular Higdon ABC, with advantages with respect to implementation and computational cost.…”

“…As a matter of fact in the original problem the atmosphere layer is an unbounded layer. What motivates this model is that, according to [14], it is possible to approximate the solution in the unbounded domain with any set precision tolerance if one applies an ABC of order high enough. At this stage, therefore, the focus is on the approximation of this problem with an absorbing boundary, rather than the physical problem with an unbounded layer.…”

Time exponential splitting technique for the Klein–Gordon equation with Hagstrom–Warburton high-order absorbing boundary conditions

“…The ABCs are built in order to achieve, after the discretization, a stable, accurate, efficient and easy to implement scheme. There exists a wide literature on this subject, see the works [5,6,10,13,14,15,16,17] and the review papers [9,11,12,26].…”

“…The parameters a j have to be chosen. We consider a j = 1 for all j, following the recommendation done in [8] and the remark in [13] in which the authors find this choice satisfactory in general. Other choices of the parameters could be considered, see for example [17] and Section 6 in this paper.…”