Ph.D Student | Tsabary Guy |
---|---|

Subject | An Iterative Solution for the Helmholtz Equation above a Rough Surface |

Department | Department of Applied Mathematics |

Supervisor | Professor Yehuda Agnon |

The essence of this
thesis report is a development and inspection of two iterative solutions of the
Helmholtz equation for a scalar field in *R*^{3} above a rough
surface that admits the Dirichlet boundary condition. The bases for the two
iterative methods are two different boundary integral equations that represent
the (unique) solution. The first integral equation is classified as a Fredholm
integral equation of the first kind. The second is classified as a Fredholm
integral equation of the second kind. This classification suggests that it is
easier to find stable solution methods to the second equation. In both methods,
we separated the boundary integral into a major part which is easy to calculate
and a local residual part. The major part is a convolution and thus can be
calculated using FFT in complexity O(N log N), where N is the number of surface
points in which the surface height and its first derivatives together with the
incoming wave and its normal derivative are all known. The residual element of
the equations can be approximated efficiently and cheaply only in surfaces
where their amplitude is less than the wavelength of the incoming wave. The iterative schemes were tested numerically against
a reference solution using Matlab computer programs that examined the
applicability range, the error estimation and the stability of the schemes. All
tests supported the supremacy of the second method. In particular the error
estimation and stability tests indicated good performance for surfaces with
slope up to 1.