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

E. H. Twizell, Abba Gumel, Q. Cao

Research output: Contribution to journalArticle

56 Citations (Scopus)

Abstract

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).

Original languageEnglish (US)
Pages (from-to)297-316
Number of pages20
JournalJournal of Mathematical Chemistry
Volume26
Issue number4
StatePublished - 2000
Externally publishedYes

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Initial value problems
Laser modes
Reaction-diffusion System
Partial differential equations
Linear systems
Nonlinear systems
Physics
Plasmas
Lasers
Nonlinear Reaction-diffusion Equations
Nonlinear Partial Differential Equations
Mathematical Modeling
Initial Value Problem
Plasma
Nonlinear Systems
Fixed point
Numerical Solution
Laser
First-order
Converge

ASJC Scopus subject areas

  • Chemistry(all)
  • Applied Mathematics

Cite this

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

In: Journal of Mathematical Chemistry, Vol. 26, No. 4, 2000, p. 297-316.

Research output: Contribution to journalArticle

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abstract = "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).",
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