@inbook{7805b34d248c4974b154bff94b90f79d,

title = "Combining the gap-tooth scheme with projective integration: Patch dynamics",

abstract = "An important class of problems exhibits macroscopically smooth behaviour in space and time, while only a microscopic evolution law is known, which describes effects on fine space and time scales. A simulation of the full microscopic problem in the whole space-time domain can therefore be prohibitively expensive. In the absence of a simplified model, we can approximate the macroscopic behaviour by performing appropriately initialized simulations of the available microscopic model in a number of small spatial domains ({"}boxes{"}) over a relatively short time interval. Here, we show how to obtain such a scheme, called {"}patch dynamics,{"} by combining the gap-tooth scheme with projective integration. The gap-tooth scheme approximates the evolution of an unavailable (in closed form) macroscopic equation in a macroscopic domain using simulations of the available microscopic model in a number of small boxes. The projective integration scheme accelerates the simulation of a problem with multiple time scales by taking a number of small steps, followed by a large extrapolation step. We illustrate this approach for a reaction-diffusion homogenization problem, and comment on the accuracy and efficiency of the method.",

keywords = "Equation-free multiscale computation, Gap-tooth scheme, Homogenization, Patch dynamics",

author = "Giovanni Samaey and Dirk Roose and Kevrekidis, {Ioannis G.}",

year = "2005",

doi = "10.1007/3-540-26444-2_12",

language = "English (US)",

isbn = "9783540253358",

series = "Lecture Notes in Computational Science and Engineering",

publisher = "Springer Verlag",

pages = "225--239",

booktitle = "Multiscale Methods in Science and Engineering",

address = "Germany",

}