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Finite‐difference methods for solving the reaction‐diffusion equations of a simple isothermal chemical system
Author(s) -
Twizell E. H.,
Wang Yigong,
Price W. G.,
Fakhr F.
Publication year - 1994
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.1690100404
Subject(s) - numerical partial differential equations , mathematics , explicit and implicit methods , nonlinear system , exponential integrator , partial differential equation , numerical stability , euler equations , numerical analysis , euler method , backward euler method , numerical methods for ordinary differential equations , ordinary differential equation , algebraic equation , multigrid method , method of lines , backward differentiation formula , reaction–diffusion system , mathematical analysis , differential equation , differential algebraic equation , physics , quantum mechanics
Numerical methods are proposed for the numerical solution of a system of reaction‐diffusion equations, which model chemical wave propagation. The reaction terms in this system of partial differential equations contain nonlinear expressions. Nevertheless, it is seen that the numerical solution is obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic system, which is often required when integrating nonlinear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations, which model the well‐stirred analogue of the chemical system. The first‐order numerical methods proposed for the solution of this initialvalue problem are characterized to be implicit. However, in each case it is seen that the numerical solution is obtained explicitly . In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well‐known, first‐order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of the partial differential equations is seen to lead to two economical and reliable methods, one sequential and one parallel, for solving the travelling‐wave problem. © 1994 John Wiley & Sons, Inc.