A conservative, spatially continuous method of lines for one-dimensional reaction-diffusion equations
International Journal of Numerical Methods for Heat & Fluid Flow
ISSN: 0961-5539
Article publication date: 6 November 2017
Abstract
Purpose
The purpose of this paper is to develop a new finite-volume method of lines for one-dimensional reaction-diffusion equations that provides piece-wise analytical solutions in space and is conservative, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficient on wave propagation.
Design/methodology/approach
A conservative, finite-volume method of lines based on piecewise integration of the diffusion operator that provides a globally continuous approximate solution and is second-order accurate is presented. Numerical experiments that assess the accuracy of the method and the time required to achieve steady state, and the effects of the nonlinear diffusion coefficients on wave propagation and boundary values are reported.
Findings
The finite-volume method of lines presented here involves the nodal values and their first-order time derivatives at three adjacent grid points, is linearly stable for a first-order accurate Euler’s backward discretization of the time derivative and has a smaller amplification factor than a second-order accurate three-point centered discretization of the second-order spatial derivative. For a system of two nonlinearly-coupled, one-dimensional reaction-diffusion equations, the amplitude, speed and separation of wave fronts are found to be strong functions of the dependence of the nonlinear diffusion coefficients on the concentration and temperature.
Originality/value
A new finite-volume method of lines for one-dimensional reaction-diffusion equations based on piecewise analytical integration of the diffusion operator and the continuity of the dependent variables and their fluxes at the cell boundaries is presented. The method may be used to study heat and mass transfer in layered media.
Keywords
Acknowledgements
The author is grateful to the referees for their comments and suggestions.
Conflict of interests – The author declares that there is no conflict of interests regarding the publication of this article.
Citation
Ramos, J.I. (2017), "A conservative, spatially continuous method of lines for one-dimensional reaction-diffusion equations", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 27 No. 11, pp. 2650-2678. https://doi.org/10.1108/HFF-12-2016-0483
Publisher
:Emerald Publishing Limited
Copyright © 2017, Emerald Publishing Limited