### 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 |
---|---|

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 |

### Fingerprint

### 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

**Numerical methods for a coupled system of differential equations arising from a thermal ignition problem.** / Pao, C. V.; Chang, Yu-Hsien; Jau, Guo Chin.

Research output: Contribution to journal › Article

*Numerical Methods for Partial Differential Equations*, vol. 29, no. 1, pp. 251-279. https://doi.org/10.1002/num.21708

}

TY - JOUR

T1 - Numerical methods for a coupled system of differential equations arising from a thermal ignition problem

AU - Pao, C. V.

AU - Chang, Yu-Hsien

AU - Jau, Guo Chin

PY - 2013/1/1

Y1 - 2013/1/1

N2 - 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

AB - 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

KW - blow-up solution

KW - catalytic converter

KW - finite difference solution

KW - monotone iteration

KW - system of first-order equations

UR - http://www.scopus.com/inward/record.url?scp=84870359347&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84870359347&partnerID=8YFLogxK

U2 - 10.1002/num.21708

DO - 10.1002/num.21708

M3 - Article

VL - 29

SP - 251

EP - 279

JO - Numerical Methods for Partial Differential Equations

JF - Numerical Methods for Partial Differential Equations

SN - 0749-159X

IS - 1

ER -