Numerical solution of problems on unbounded domains. A review

While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABCs) at the newly formed external boundary. The issue of setting the ABCs appears most significant in many areas of scientific computing, for example, in problems originating from acoustics, electrodynamics, solid mechanics, and fluid dynamics. 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 overall quality and performance of numerical algorithms and interpretation of the results. Most of the currently used techniques for setting the ABCs can basically be classified into two groups. The methods from the first group (global ABCs) usually provide high accuracy and robustness of the numerical procedure but often appear to be fairly cumbersome 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 computations. In this paper we first present an extensive survey and provide a comparative assessment of different existing methods for constructing the ABCs. Then, we describe a new ABCs technique proposed in our recent work and review the corresponding results. This new technique enables one to construct the ABCs that largely combine the advantages relevant to the two aforementioned classes of existing methods. Our approach is based on application of the difference potentials method by Ryaben'kii. This approach allows one to obtain highly accurate ABCs in the form of certain (nonlocal) boundary operator equations. The operators involved are analogous to the pseudodifferential boundary projections first introduced by Calderon and then also studied by Seeley. In spite of the nonlocality, the new boundary conditions are geometrically universal, numerically inexpensive, and easy to implement along with the existing solvers, (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.