Artificial boundary conditions of absolute transparency for two- and three-dimensional external time-dependent scattering problems

Abstract

For an external problem in IRd (d=2, 3) such that the unknown function satisfies the wave equation outside a finite domain, we generate artificial boundary conditions transparent to outgoing waves. These conditions permit an equivalent replacement of the original external problem by the problem inside the artificial boundary which is a circle (d=2) or a sphere (d=3): The questions of numerical implementation of the artificial conditions (that are non-local in both space and time) are considered. Special attention is paid to the reduction of necessary computational resources; in particular, a way of incorporating these conditions into numerical methods which makes the computational formulae local in time is suggested. The aspects of treating artificial boundaries of a non-spherical shape are discussed. Numerical examples of two- and three-dimensional scattering problems demonstrate the accuracy of proposed artificial boundary conditions.

Radiation Boundary Conditions for Maxwell's Equations: A Review of Accurate Time-Domain Formulations

“…That is, one can avoid the storage of the time history evident in (2.33)-(2.34). This fact was independently exploited by Sofronov [63,64] and Grote and Keller [28,29,30] to derive and implement accurate temporally local conditions. (See also [31] for applications to multiple scattering.)…”

“…For the case of a spherical boundary, we have already observed that the kernels are exactly equal to sums of exponentials (2.35). This leads to an algorithm which is mathematically equivalent to the one proposed by Sofronov [63,64] and Grote-Keller [28,29,30]:…”

Exact nonreflecting boundary conditions for three dimensional poroelastic wave equations

“…This successful method is then applied to Maxwell's equations [30] and elastic waves [31]. Another form of the exact nonreflecting boundary conditions for the scalar wave equation was obtained by Sofronov [54] independently. A systematic approach to the computation of exact conditions has been studied [1].…”

High‐order Higdon‐like boundary conditions for exterior transient wave problems

“…They extended this NRBC for the case of elastic waves in Reference [13]. Sofronov [14,15] has independently published a similar scheme in the Russian literature. Hagstrom and Hariharan [16,17] constructed high-order NRBCs for the two-and three-dimensional time-dependent wave equations based on the analytic series representation for the outgoing solutions of these equations.…”