| Home > Publications > Reports > Numerical Analysis and Applied Mathematics (TW) |
TW 370
Giovanni Samaey, Ioannis G. Kevrekidis and Dirk Roose
Damping factors for the gap-tooth scheme
Abstract
An important class of problems exhibits macroscopically smooth behaviour
in space and time, while only a microscopic evolution law is known.
For such time-dependent multi-scale problems, the gap-tooth scheme has
recently been proposed.
The scheme approximates the evolution of an unavailable (in closed form)
macroscopic equation in a macroscopic domain; it only uses appropriately
initialized simulations
of the available microscopic model in a number of small boxes.
For some model problems, including numerical homogenization, the scheme
is essentially equivalent to a finite difference scheme, provided we
repeatedly impose appropriate algebraic constraints on the solution for
each box.
Here, we demonstrate that it is possible to obtain a convergent scheme
without constraining the microscopic code, by introducing buffers that
"shield" over relatively short times the dynamics
inside each box from boundary effects.
We explore and quantify the behavior of these schemes
systematically through the numerical computation of damping factors
of the corresponding coarse time-stepper, for which no closed formula is
available.
