A fast boundary element algorithm for time-dependant potential problems

The numerical solution of time-dependant potential problems via the boundary element has been crippled by the high computational cost due to the inherent time history constraint in the integral representation. Using a boundary-only formulation, the time integrations, at any instant in time, have to be evaluated starting from the initial time. This time-history dependence becomes impractical and inadequate for problems where computations are to be performed for extended times. This also made the boundary element uncompetitive compared to the domain-mesh based methods, such as finite difference and finite element methods, for the solution of transient potential problems. Generally, the evaluation of the potential at N domain points using M boundary points at the Kth time step requires an amount of computer operations of the order O(KM²+KNM). This paper presents an algorithm which requires a computational cost of the order of only O(M²+NM), where the dependence from the past K-steps is removed. The algorithm combines the boundary element method and a scheme, which uses virtual collocation points and radial basis functions to approximate the domain integral.