Nonreflecting boundary conditions for time-dependent wave propagation

Many problems in computational science arise in unbounded domains and
thus require an artificial boundary B, which truncates the unbounded exterior
domain and restricts the region of interest to a finite computational
domain,
. It then becomes necessary to impose a boundary condition at
B, which ensures that the solution in
coincides with the restriction to
of the solution in the unbounded region. If we exhibit a boundary condition,
such that the fictitious boundary appears perfectly transparent, we shall call
it exact. Otherwise it will correspond to an approximate boundary condition
and generate some spurious reflection, which travels back and spoils the
solution everywhere in the computational domain. In addition to the transparency
property, we require the computational effort involved with such a
boundary condition to be comparable to that of the numerical method used
in the interior. Otherwise the boundary condition will quickly be dismissed
as prohibitively expensive and impractical. The constant demand for increasingly
accurate, efficient, and robust numerical methods, which can handle a
wide variety of physical phenomena, spurs the search for improvements in
artificial boundary conditions.
In the last decade, the perfectly matched layer (PML) approach [16] has
proved a flexible and accurate method for the simulation of waves in unbounded
media. Standard PML formulations, however, usually require wave
equations stated in their standard second-order form to be reformulated as
first-order systems, thereby introducing many additional unknowns. To circumvent
this cumbersome and somewhat expensive step we propose instead
a simple PML formulation directly for the wave equation in its second-order
form. Our formulation requires fewer auxiliary unknowns than previous formulations
[23, 94].
Starting from a high-order local nonreflecting boundary condition (NRBC)
for single scattering [55], we derive a local NRBC for time-dependent multiple
scattering problems, which is completely local both in space and time. To do so, we first develop a high order exterior evaluation formula for a purely
outgoing wave field, given its values and those of certain auxiliary functions needed for the local NRBC on the artificial boundary. By combining that
evaluation formula with the decomposition of the total scattered field into
purely outgoing contributions, we obtain the first exact, completely local,
NRBC for time-dependent multiple scattering. Remarkably, the information
transfer (of time retarded values) between sub-domains will only occur
across those parts of the artificial boundary, where outgoing rays intersect
neighboring sub-domains, i.e. typically only across a fraction of the artificial
boundary. The accuracy, stability and efficiency of this new local NRBC is
evaluated by coupling it to standard finite element or finite difference methods.