### 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 language | English (US) |
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Title of host publication | Multiscale Methods in Science and Engineering |

Publisher | Springer Verlag |

Pages | 225-239 |

Number of pages | 15 |

ISBN (Print) | 9783540253358 |

DOIs | |

State | Published - Jan 1 2005 |

### Publication series

Name | Lecture Notes in Computational Science and Engineering |
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Volume | 44 |

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

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## Cite this

*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