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

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.

Original languageEnglish (US)
Title of host publicationMultiscale Methods in Science and Engineering
PublisherSpringer Verlag
Pages225-239
Number of pages15
ISBN (Print)9783540253358
DOIs
StatePublished - Jan 1 2005

Publication series

NameLecture Notes in Computational Science and Engineering
Volume44
ISSN (Print)1439-7358

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Engineering(all)
  • Discrete Mathematics and Combinatorics
  • Control and Optimization
  • Computational Mathematics

Keywords

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

Fingerprint Dive into the research topics of 'Combining the gap-tooth scheme with projective integration: Patch dynamics'. Together they form a unique fingerprint.

  • Cite this

    Samaey, G., Roose, D., & Kevrekidis, I. G. (2005). Combining the gap-tooth scheme with projective integration: Patch dynamics. In Multiscale Methods in Science and Engineering (pp. 225-239). (Lecture Notes in Computational Science and Engineering; Vol. 44). Springer Verlag. https://doi.org/10.1007/3-540-26444-2_12