### Abstract

We present an "equation-free" multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast in the form of stochastic differential equations. A detailed fine-scale computation of such a problem requires discretization and solution of a large system of equations and can be prohibitively time consuming. To circumvent this difficulty, we propose an equation-free approach, where the fine-scale computation is conducted only for a (small) fraction of the overall time. The evolution of a set of appropriately defined coarse-grained variables (observables) is evaluated during the fine-scale computation, and "projective integration" is used to accelerate the integration. The choice of these coarse variables is an important part of the approach: they are the coefficients of pointwise polynomial expansions of the random solutions. Such a choice of coarse variables allows us to reconstruct representative ensembles of fine-scale solutions with "correct" correlation structures, which is a key to algorithm efficiency. Numerical examples demonstrating accuracy and efficiency of the approach are presented.

Original language | English (US) |
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Pages (from-to) | 915-935 |

Number of pages | 21 |

Journal | Multiscale Modeling and Simulation |

Volume | 4 |

Issue number | 3 |

DOIs | |

State | Published - 2005 |

### All Science Journal Classification (ASJC) codes

- Chemistry(all)
- Modeling and Simulation
- Ecological Modeling
- Physics and Astronomy(all)
- Computer Science Applications

### Keywords

- Diffusion in random media
- Equation-free
- Multiscale problem
- Stochastic modeling

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## Cite this

*Multiscale Modeling and Simulation*,

*4*(3), 915-935. https://doi.org/10.1137/040615006