Exact non-reflecting boundary condition for 3D time-dependent multiple scattering–multiple source problems

Abstract

We consider some 3D wave equation problems defined in an unbounded domain, possibly with far field sources. For their solution, by means of standard finite element methods, we propose a Non Reflecting Boundary Condition (NRBC) on the chosen artificial boundary B, which is based on a known space–time integral equation defining a relationship between the solution of the differential problem and its normal derivative on B. Such a NRBC is exact, non local both in space and time. We discretize it by using a fast convolution quadrature technique in time and a collocation method in space. The computational complexity of the discrete convolution is of order NlogN, being N the total number of time steps performed. That of the fully discretized NRBC is O(NB2NlogN), where NB denotes the number of mesh points taken on B.Besides showing a good accuracy and numerical stability, the proposed NRBC has the property of being suitable for artificial boundaries of general shapes. It also allows the treatment of far field (multiple) sources, that do not have to be necessarily included in the finite computational domain, being transparent not only for outgoing waves but also for incoming ones. This approach is in particular used to solve multiple scattering problems.