### Abstract

Original language | English |
---|---|

Pages (from-to) | 757-777 |

Number of pages | 21 |

Journal | Journal of Computational Physics |

Volume | 215 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jul 2006 |

### Fingerprint

### Keywords

- Nonlinear reaction-diffusion equations
- Quenching singularity
- Moving mesh method
- Peaceman–Rachford splitting
- ADI monotonicity
- Parallelization

### Cite this

*Journal of Computational Physics*,

*215*(2), 757-777. https://doi.org/10.1016/j.jcp.2005.11.019

}

*Journal of Computational Physics*, vol. 215, no. 2, pp. 757-777. https://doi.org/10.1016/j.jcp.2005.11.019

**A splitting moving mesh method for reaction-diffusion equations of quenching type.** / Liang, Kewei; Lin, Ping; Ong, Ming Tze; Tan, Roger C. E.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A splitting moving mesh method for reaction-diffusion equations of quenching type

AU - Liang, Kewei

AU - Lin, Ping

AU - Ong, Ming Tze

AU - Tan, Roger C. E.

N1 - dc.publisher: Elsevier

PY - 2006/7

Y1 - 2006/7

N2 - This paper studies the numerical solution of multi-dimensional nonlinear degenerate reaction–diffusion differential equations with a singular force term over a rectangular domain. The equations may generate strong quenching singularities. Our work focuses on a variable temporal step Peaceman–Rachford splitting method with an adaptive moving mesh in space. The temporal and spatial adaptation is implemented based on arc-length type of estimations of the time derivative of the solution since the time derivative of the solution approaches infinity when the quenching occurs. The multi-dimensional problem is split into a few one-dimensional problems and the splitting procedure can also be parallelized so that the computational time is significantly reduced. The physical monotonicity of the solution and stability of this variable step moving mesh scheme are analyzed for the time away from the quenching. As stability analysis may not be valid when it is very close to the quenching, thus an exact linear problem is introduced to justify the stability near the quenching time. Finally we provide some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method for the quenching problem or other problems with point singularities. We will also show the significant reduction in computational time required with parallel implementation of the algorithm on a computer with multi-CPU.

AB - This paper studies the numerical solution of multi-dimensional nonlinear degenerate reaction–diffusion differential equations with a singular force term over a rectangular domain. The equations may generate strong quenching singularities. Our work focuses on a variable temporal step Peaceman–Rachford splitting method with an adaptive moving mesh in space. The temporal and spatial adaptation is implemented based on arc-length type of estimations of the time derivative of the solution since the time derivative of the solution approaches infinity when the quenching occurs. The multi-dimensional problem is split into a few one-dimensional problems and the splitting procedure can also be parallelized so that the computational time is significantly reduced. The physical monotonicity of the solution and stability of this variable step moving mesh scheme are analyzed for the time away from the quenching. As stability analysis may not be valid when it is very close to the quenching, thus an exact linear problem is introduced to justify the stability near the quenching time. Finally we provide some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method for the quenching problem or other problems with point singularities. We will also show the significant reduction in computational time required with parallel implementation of the algorithm on a computer with multi-CPU.

KW - Nonlinear reaction-diffusion equations

KW - Quenching singularity

KW - Moving mesh method

KW - Peaceman–Rachford splitting

KW - ADI monotonicity

KW - Parallelization

U2 - 10.1016/j.jcp.2005.11.019

DO - 10.1016/j.jcp.2005.11.019

M3 - Article

VL - 215

SP - 757

EP - 777

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 2

ER -