At the boundary of infinity: numerical methods for unbounded domains (Q1570889)

The paper is the revised text of the author's inaugural lecture, given in April 1998 at the ETH Zürich. It is devoted to the numerical analysis of physical problems in unbounded domains. The key idea for a solution of such a problem, e.g. the wave propagation in one or more dimensions, is the introduction of an artificial boundary and appropriate conditions on it. For the convergence of an approximate solution to the exact one, the discretization error as well as the error due to the artificial boundary have to be studied. In addition, an approximation has to be efficient, as the author shows by a simple example. The necessary steps for the solution of an unbounded domain problem are discussed for the one-dimensional linear wave equation. Afterwards, the author turns to the much more complicated multidimensional case, and pseudo-differential operators come into play in view of nonlocal boundary conditions. Alternatively, the author presents the idea of Kirchhoff's integral representation that is based upon the famous Huygens's principle. Finally, numerical tests illustrate different approaches and the inherent question of convergence. The paper gives quite a nice insight into this highly interesting but complicated field of research and is addressed to the non-specialist.