  In recent years, individual-based/agent-based modeling has been applied to
study a wide range of applications, ranging from engineering problems to
phenomena in sociology, economics and biology. Simulating such agent-based
models over extended spatiotemporal domains can be prohibitively expensive due
to stochasticity and the presence of multiple scales. Nevertheless, many
agent-based problems exhibit smooth behavior in space and time on a macroscopic
scale, suggesting that a useful coarse-grained continuum model could be
obtained. For such problems, the equation-free framework [16-18] can
significantly reduce the computational cost. Patch dynamics is an essential
component of this framework. This scheme is designed to perform numerical
simulations of an unavailable macroscopic equation on macroscopic time and
length scales; it uses appropriately initialized simulations of the fine-scale
agent-based model in a number of small "patches", which cover only a fraction
of the spatiotemporal domain. In this work, we construct a
finite-volume-inspired conservative patch dynamics scheme and apply it to a
financial market agent-based model based on the work of Omurtag and Sirovich
[22]. We first apply our patch dynamics scheme to a continuum approximation of
the agent-based model, to study its performance and analyze its accuracy. We
then apply the scheme to the agent-based model itself. Our computational
experiments indicate that here, typically, the patch dynamics-based simulation
requires only 20% of the full agent-based simulation in space, and need occur
over only 10% of the temporal domain.
