### Abstract

We present a splitting moving mesh method for multi-dimensional reaction-diffusion problems with nonlinear forcing terms over rectangular domains.The structure of the adaptive algorithm is an elegant combination of an operator splitting and one-dimensional moving mesh. It is motivated by the nature of splitting method, which splits a multi-dimensional problems into a few one-dimensional problems. Therefore, the temporal and the spatial adaptations are adopted based on one-dimensional arc-length equidistributed rule. The method not only keeps the advantage of splitting methods on reducing computational cost but also simplifies the implementation of moving mesh techniques. The method is first presented in [20] for the 2-D quenching problems using Peaceman-Rachford splitting. In this paper, we apply the splitting moving mesh method to both the 3-D blow-up and the 3-D quenching singularity using Douglas splitting. 3-D numerical examples will be given to demonstrate the good performance of the method.

Original language | English |
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Title of host publication | Recent advances in adaptive computation: proceedings of the International Conference on Recent Advances in Adaptive Computation, Hangzhou, China, May 24-28, 2004 |

Editors | Z.-C. Shi, Z. Chen, T. Tang, D. Yu |

Publisher | American Mathematical Society |

Pages | 311-324 |

Number of pages | 14 |

ISBN (Print) | 9780821836620 |

Publication status | Published - 2005 |

### Publication series

Name | Contemporary mathematics |
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Number | 383 |

### Keywords

- Reaction-diffusion
- Splitting methods

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## Cite this

Liang, K., & Lin, P. (2005). A splitting moving mesh method for 3-D quenching and blow-up problems. In Z-C. Shi, Z. Chen, T. Tang, & D. Yu (Eds.),

*Recent advances in adaptive computation: proceedings of the International Conference on Recent Advances in Adaptive Computation, Hangzhou, China, May 24-28, 2004*(pp. 311-324). (Contemporary mathematics; No. 383). American Mathematical Society. http://copac.ac.uk/crn/72013413396/html