### Abstract

This article is concerned with monotone iterative methods for numerical solutions of a coupled system of a first-order partial differential equation and an ordinary differential equation which arises from fast-igniting catalytic converters in automobile engineering. The monotone iterative scheme yields a straightforward marching process for the corresponding discrete system by the finite-difference method, and it gives not only a computational algorithm for numerical solutions of the problem but also the existence and uniqueness of a finite-difference solution. Particular attention is given to the "finite-time" blow-up property of the solution. In terms of minimal sequence of the monotone iterations, some necessary and sufficient conditions for the blow-up solution are obtained. Also given is the convergence of the finite-difference solution to the continuous solution as the mesh size tends to zero. Numerical results of the problem, including a case where the continuous solution is explicitly known, are presented and are compared with the known solution. Special attention is devoted to the computation of the blow-up time and the critical value of a physical parameter which determines the global existence and the blow-up property of the solution. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Original language | English |
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Pages (from-to) | 251-279 |

Number of pages | 29 |

Journal | Numerical Methods for Partial Differential Equations |

Volume | 29 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 Jan 1 |

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### Keywords

- blow-up solution
- catalytic converter
- finite difference solution
- monotone iteration
- system of first-order equations

### ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*Numerical Methods for Partial Differential Equations*,

*29*(1), 251-279. https://doi.org/10.1002/num.21708