Combining the gap-tooth scheme with projective integration: Patch dynamics

Giovanni Samaey, Dirk Roose, Ioannis G. Kevrekidis

Research output: Chapter in Book/Report/Conference proceedingChapter

4 Scopus citations


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.

Original languageEnglish (US)
Title of host publicationMultiscale Methods in Science and Engineering
PublisherSpringer Verlag
Number of pages15
ISBN (Print)9783540253358
StatePublished - Jan 1 2005

Publication series

NameLecture Notes in Computational Science and Engineering
ISSN (Print)1439-7358

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • General Engineering
  • Discrete Mathematics and Combinatorics
  • Control and Optimization
  • Computational Mathematics


  • Equation-free multiscale computation
  • Gap-tooth scheme
  • Homogenization
  • Patch dynamics


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