PMAA2012 ABSTRACT SUBMISSION

Presenting author: Dan Gordon 

Co-authors: Eli Turkel, Rachel Gordon and Semyon Tsynkov 

Abstract number: 18

TITLE: Parallel implementation of compact sixth order schemes 
for the Helmholtz equation with variable wave number

ABSTRACT:
Solving the Helmholtz equation at high frequencies is notoriously 
difficult.  The problem is compounded by the ``pollution effect'', 
according to which the required number of intervals per domain side 
is proportional to $k^{(p+1)/p}$, where $k$ is the wave number and 
$p$ is the order of accuracy of the finite difference discretization 
scheme.  Hence, high order schemes have a clear advantage with large 
wave numbers.  Ideally, the high order schemes should be compact, 
meaning that they require only a 9-point stencil in 2D and and a 
27-point stencil in 3D.  For many practical problems in various areas 
such as geophysics, the problems are huge and three-dimensional, so 
direct methods are not viable.  Furthermore, such problems are usually 
set in heterogeneous domains, resulting in a variable wave number. 
This work presents compact 2D and 3D sixth order schemes for the 
Helmholtz equation with variable wave number.  The new schemes were 
tested on 2D and 3D problems with variable wave numbers and known 
analytic solutions, showing that at high frequencies, the new schemes 
were far superior to the 2nd order scheme.  Implemented with the 
parallel CARP-CG algorithm, the compact schemes do not require 
more inter-processor communications than the 2nd order schemes.