Waveform Relaxation for Differential-Functional Equations
Abstract
We present results on the convergence of waveform relaxation
(WR) techniques, in particular new error estimates for equations with
time-dependent coefficients. New conclusions about the relationship
between the convergence of WR and the coefficients in
differential-functional equation will be derived. The convergence of
the waveform method and the quality of the a priori error bounds will
be illustrated by means of extensive numerical data obtained for
parabolic differential-functional equations. We
compare the WR convergence after the application of the finite
difference (FD)
semi-discretization and the Chebyshev pseudospectral (ChPS)
semi-discretization. A
result on the convergence of the ChPS semi-discretization
for parabolic functional
equations will be presented as well. Our conclusion, valid for both
parabolic differential and differential-functional equations, is
that, although differentiation matrices for ChPS are dense, while
differentiation matrices originating in FD are sparse, WR error
bounds and WR convergence are better in the first case. Moreover, for
the same accuracy of semi-discretization, a single Gauss-Seidel WR
iteration for ChPS is less expensive than in the case of FD.
