Construction of artificial boundary conditions using difference potentials method

One of the widely used techniques for the numerical treatment of infinite-domain problems is based on implementation of the so-called artificial boundary conditions (ABC's). These boundary conditions are typically set at the external boundary of a finite computational domain once the latter is obtained from the original unbounded domain by means of truncation. Since the infinite-domain problems present a wide class of practically important formulations, for example, in solid and fluid mechanics, electrodynamics, and acoustics, the issue of constructing and implementing the ABC's becomes most significant in many areas of computational mathematics. In this paper, we review a fairly recent approach to constructing the ABC's. This approach, which is based on the difference potentials method (DPM), enables one to achieve high accuracy of computations and robustness of the numerical procedure and at the same time provides for a geometrically universal and algorithmically simple technique. The DPM-based approach has already demonstrated its explicit superiority over other existing methods in many important cases and we expect it to become one of the powerful tools in modern scientific computing.