@TechReport{ it:2004-042,
author = {Per Sundqvist},
title = {Boundary Summation Equations},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2004},
number = {2004-042},
month = sep,
abstract = {A new solution method for systems of partial difference
equations is presented. It can be seen as a discrete
counterpart of boundary integral equations, but with sums
instead of integrals. The number of unknowns in systems of
linear difference equations with constant coefficients
defined on uniform $d$-dimensional grids are reduced so
that one dimension is eliminated.
The reduction is obtained using fundamental solutions of
difference operators, yielding a reduced system that is
dense. The storage of the reduced system requires
$\mathcal{O}(N)$ memory positions, where $N$ is the length
of the original vector of unknowns. The application of the
matrix utilizes fast Fourier transform as its most complex
operation, and requires hence $\mathcal{O}(N\log N)$
arithmetic operations.
Numerical experiments are performed, exploring the behavior
of GMRES when applied to reduced systems originating from
discretizations of partial differential equations. Model
problems are chosen to include scalar equations as well as
systems, with various boundary conditions, and on
differently shaped domains. The new solution method
performs well for an upwind discretization of an inviscid
flow-problem.
A proof of grid independent convergence is given for a
simpler iterative method applied to a specific
discretization of a first order differential equation. The
numerical experiments indicate that this property carries
over to many other problems in the same class. }
}