17 research outputs found
Efficient PML for the wave equation
In the last decade, the perfectly matched layer (PML) approach has proved a
flexible and accurate method for the simulation of waves in unbounded media.
Most 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 instead propose a simple PML formulation directly
for the wave equation in its second-order form. Inside the absorbing layer, our
formulation requires only two auxiliary variables in two space dimensions and
four auxiliary variables in three space dimensions; hence it is cheap to
implement. Since our formulation requires no higher derivatives, it is also
easily coupled with standard finite difference or finite element methods.
Strong stability is proved while numerical examples in two and three space
dimensions illustrate the accuracy and long time stability of our PML
formulation.Comment: 16 pages, 6 figure
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
On local nonreflecting boundary conditions for time dependent wave propagation
The simulation of wave phenomena in unbounded domains generally requires an artificial boundary to truncate the unbounded exterior and limit the computation to a finite region. At the artificial boundary a boundary condition is then needed, which allows the propagating waves to exit the computational domain without spurious reflection. In 1977, Engquist and Majda proposed the first hierarchy of absorbing boundary conditions, which allows a systematic reduction of spurious reflection without moving the artificial boundary farther away from the scatterer. Their pioneering work, which initiated an entire research area, is reviewed here from a modern perspective. Recent developments such as high-order local conditions and their extension to multiple scattering are also presented. Finally, the accuracy of high-order local conditions is demonstrated through numerical experiment