### Abstract

Spatially two-dimensional, possibly degenerate reaction-diffusion systems, with a focus on models of combustion, pattern formation and chemotaxis, are solved by a fully adaptive multiresolution scheme. Solutions of these equations exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. This calls for a concentration of computational effort on zones of strong variation. The multiresolution scheme is based on finite volume discretizations with explicit time stepping. The multiresolution representation of the solution is stored in a graded tree ("quadtree"), whose leaves are the non-uniform finite volumes on whose borders the numerical divergence is evaluated. By a thresholding procedure, namely the elimination of leaves with solution values that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen such that the total error of the adaptive scheme is of the same order as that of the reference finite volume scheme. Since chemical reactions involve a large range of temporal scales, but are spatially well localized (especially in the combustion model), a locally varying adaptive time stepping strategy is applied. For scalar equations, this strategy has the advantage that consistence with a CFL condition is always enforced. Numerical experiments with five different scenarios, in part with local time stepping, illustrate the effectiveness of the adaptive multiresolution method. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled.

Original language | English |
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Pages (from-to) | 1668-1692 |

Number of pages | 25 |

Journal | Applied Numerical Mathematics |

Volume | 59 |

Issue number | 7 |

DOIs | |

Publication status | Published - 1 Jul 2009 |

Externally published | Yes |

### Keywords

- Adaptive multiresolution scheme
- Chemotaxis
- Degenerate parabolic equation
- Finite volume schemes
- Flame balls interaction
- Keller-Segel systems
- Locally varying time stepping
- Pattern formation

## Cite this

*Applied Numerical Mathematics*,

*59*(7), 1668-1692. https://doi.org/10.1016/j.apnum.2008.12.001