### Abstract

A second-order method is developed for the numerical solution of the initial-value problems ′ ≡ du/dt = f_{1}(u, v), t > 0, u(0) = U^{0} and v′ ≡ dv/dt = f_{2}(u, v), t > 0, v(0) = V^{0}, in which the functions f_{1}(u, v) = B + u^{2}v - (A + 1)u and f_{2}(u, v) = Au - u^{2}v, where A and B are positive real constants, are the reaction terms arising from the mathematical modelling of chemical systems such as in enzymatic reactions and plasma and laser physics in multiple coupling between modes. The method is based on three first-order methods for solving u and v, respectively. In addition to being second-order accurate in space and time, the method is seen to converge to the correct fixed point (U^{*} = B, V^{*} = A/B) provided 1 - A + B^{2} ≥ 0. The approach adopted is extended to solve a class of non-linear reaction-diffusion equations in two-space dimensions known as the "Brusselator" system. The algorithm is implemented in parallel using two processors, each solving a linear algebraic system as opposed to solving non-linear systems, which is often required when integrating non-linear partial differential equations (PDEs).

Original language | English (US) |
---|---|

Pages (from-to) | 297-316 |

Number of pages | 20 |

Journal | Journal of Mathematical Chemistry |

Volume | 26 |

Issue number | 4 |

State | Published - 2000 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Chemistry(all)
- Applied Mathematics

### Cite this

*Journal of Mathematical Chemistry*,

*26*(4), 297-316.

**A second-order scheme for the "Brusselator" reaction-diffusion system.** / Twizell, E. H.; Gumel, Abba; Cao, Q.

Research output: Contribution to journal › Article

*Journal of Mathematical Chemistry*, vol. 26, no. 4, pp. 297-316.

}

TY - JOUR

T1 - A second-order scheme for the "Brusselator" reaction-diffusion system

AU - Twizell, E. H.

AU - Gumel, Abba

AU - Cao, Q.

PY - 2000

Y1 - 2000

N2 - A second-order method is developed for the numerical solution of the initial-value problems ′ ≡ du/dt = f1(u, v), t > 0, u(0) = U0 and v′ ≡ dv/dt = f2(u, v), t > 0, v(0) = V0, in which the functions f1(u, v) = B + u2v - (A + 1)u and f2(u, v) = Au - u2v, where A and B are positive real constants, are the reaction terms arising from the mathematical modelling of chemical systems such as in enzymatic reactions and plasma and laser physics in multiple coupling between modes. The method is based on three first-order methods for solving u and v, respectively. In addition to being second-order accurate in space and time, the method is seen to converge to the correct fixed point (U* = B, V* = A/B) provided 1 - A + B2 ≥ 0. The approach adopted is extended to solve a class of non-linear reaction-diffusion equations in two-space dimensions known as the "Brusselator" system. The algorithm is implemented in parallel using two processors, each solving a linear algebraic system as opposed to solving non-linear systems, which is often required when integrating non-linear partial differential equations (PDEs).

AB - A second-order method is developed for the numerical solution of the initial-value problems ′ ≡ du/dt = f1(u, v), t > 0, u(0) = U0 and v′ ≡ dv/dt = f2(u, v), t > 0, v(0) = V0, in which the functions f1(u, v) = B + u2v - (A + 1)u and f2(u, v) = Au - u2v, where A and B are positive real constants, are the reaction terms arising from the mathematical modelling of chemical systems such as in enzymatic reactions and plasma and laser physics in multiple coupling between modes. The method is based on three first-order methods for solving u and v, respectively. In addition to being second-order accurate in space and time, the method is seen to converge to the correct fixed point (U* = B, V* = A/B) provided 1 - A + B2 ≥ 0. The approach adopted is extended to solve a class of non-linear reaction-diffusion equations in two-space dimensions known as the "Brusselator" system. The algorithm is implemented in parallel using two processors, each solving a linear algebraic system as opposed to solving non-linear systems, which is often required when integrating non-linear partial differential equations (PDEs).

UR - http://www.scopus.com/inward/record.url?scp=0033114412&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033114412&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033114412

VL - 26

SP - 297

EP - 316

JO - Journal of Mathematical Chemistry

JF - Journal of Mathematical Chemistry

SN - 0259-9791

IS - 4

ER -