Time-dependent problems with the boundary integral equation method

Time-dependent problems that are modeled by initial-boundary value problems for parabolic or hyperbolic partial differential equations can be treated with the boundary integral equation method. The ideal situation is when the right-hand side in the partial differential equation and the initial conditions vanish, the data are given only on the boundary of the domain, the equation has constant coefficients, and the domain does not depend on time. In this situation, the transformation of the problem to a boundary integral equation follows the same well-known lines as for the case of stationary or time-harmonic problems modeled by elliptic boundary value problems. The same main advantages of the reduction to the boundary prevail: Reduction of the dimension by one, and reduction of an unbounded exterior domain to a bounded boundary. There are, however, specific difficulties due to the additional time dimension: Apart from the practical problems of increased complexity related to the higher dimension, there can appear new stability problems. In the stationary case, one often has unconditional stability for reasonable approximation methods, and this stability is closely related to variational formulations based on the ellipticity of the underlying boundary value problem. In the timedependent case, instabilities have been observed in practice, but due to the absence of ellipticity, the stability analysis is more difficult and fewer theoretical results are available. In this article, the mathematical principles governing the construction of boundary integral equation methods for time-dependent problems are presented. We describe some of the main algorithms that are used in practice and have been analyzed in the mathematical literature.