23 research outputs found
An accurate boundary value problem solver applied to scattering from cylinders with corners
In this paper we consider the classic problems of scattering of waves from
perfectly conducting cylinders with piecewise smooth boundaries. The scattering
problems are formulated as integral equations and solved using a Nystr\"om
scheme where the corners of the cylinders are efficiently handled by a method
referred to as Recursively Compressed Inverse Preconditioning (RCIP). This
method has been very successful in treating static problems in non-smooth
domains and the present paper shows that it works equally well for the
Helmholtz equation. In the numerical examples we specialize to scattering of E-
and H-waves from a cylinder with one corner. Even at a size kd=1000, where k is
the wavenumber and d the diameter, the scheme produces at least 13 digits of
accuracy in the electric and magnetic fields everywhere outside the cylinder.Comment: 19 pages, 3 figure
A high-order Nystrom discretization scheme for boundary integral equations defined on rotationally symmetric surfaces
A scheme for rapidly and accurately computing solutions to boundary integral
equations (BIEs) on rotationally symmetric surfaces in R^3 is presented. The
scheme uses the Fourier transform to reduce the original BIE defined on a
surface to a sequence of BIEs defined on a generating curve for the surface. It
can handle loads that are not necessarily rotationally symmetric. Nystrom
discretization is used to discretize the BIEs on the generating curve. The
quadrature is a high-order Gaussian rule that is modified near the diagonal to
retain high-order accuracy for singular kernels. The reduction in
dimensionality, along with the use of high-order accurate quadratures, leads to
small linear systems that can be inverted directly via, e.g., Gaussian
elimination. This makes the scheme particularly fast in environments involving
multiple right hand sides. It is demonstrated that for BIEs associated with the
Laplace and Helmholtz equations, the kernel in the reduced equations can be
evaluated very rapidly by exploiting recursion relations for Legendre
functions. Numerical examples illustrate the performance of the scheme; in
particular, it is demonstrated that for a BIE associated with Laplace's
equation on a surface discretized using 320,800 points, the set-up phase of the
algorithm takes 1 minute on a standard laptop, and then solves can be executed
in 0.5 seconds.Comment: arXiv admin note: substantial text overlap with
arXiv:1012.56301002.200