NUMERICAL-SOLUTION OF PROBLEMS ON UNBOUNDED-DOMAINS - A REVIEW

While numerically solving a problem initially formulated on an unbound ed domain, one typically truncates this domain, which necessitates set ting the artificial boundary conditions (ABCs) at the newly formed ext ernal boundary. The issue of setting the ABCs appears most significant in many areas of scientific computing, for example, in problems origi nating from acoustics, electrodynamics, solid mechanics, and fluid dyn amics. In particular, in computational fluid dynamics (where external problems represent a wide class of important formulations) the proper treatment of external boundaries may have a profound impact on the ove rall quality and performance of numerical algorithms and interpretatio n of the results. Most of the currently used techniques for setting th e ABCs can basically be classified into two groups. The methods from t he first group (global ABCs) usually provide high accuracy and robustn ess of the numerical procedure but often appear to be fairly cumbersom e and (computationally) expensive. The methods from the second group ( local ABCs) are, as a rule, algorithmically simple, numerically cheap, and geometrically universal; however, they usually lack accuracy of c omputations. In this paper we first present an extensive survey and pr ovide a comparative assessment of different existing methods for const ructing the ABCs. Then, we describe a new ABCs technique proposed in o ur recent work and review the corresponding results. This new techniqu e enables one to construct the ABCs that largely combine the advantage s relevant to the two aforementioned classes of existing methods. Our approach is based on application of the difference potentials method b y Ryaben'kii. This approach allows one to obtain highly accurate ABCs in the form of certain (nonlocal) boundary operator equations. The ope rators involved are analogous to the pseudodifferential boundary proje ctions first introduced by Calderon and then also studied by Seeley. I n spite of the nonlocality, the new boundary conditions are geometrica lly universal, numerically inexpensive, and easy to implement along wi th the existing solvers, (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.