  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.
