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A conservative, piecewise–analytical, transversal method of lines for reaction–diffusion equations

A conservative, piecewise–analytical, transversal method of lines for reaction–diffusion equations The purpose of this paper is to develop a new transversal method of lines for one-dimensional reaction–diffusion equations that is conservative and provides piecewise–analytical solutions in space, analyze its truncation errors and linear stability, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficients, reaction rate terms and initial conditions on wave propagation and merging.Design/methodology/approachA conservative, transversal method of lines based on the discretization of time and piecewise analytical integration of the resulting two-point boundary-value problems subject to the continuity of the dependent variables and their fluxes at the control-volume boundaries, is presented. The method provides three-point finite difference expressions for the nodal values and continuous solutions in space, and its accuracy has been determined first analytically and then assessed in numerical experiments of reaction-diffusion problems, which exhibit interior and/or boundary layers.FindingsThe transversal method of lines presented here results in three-point finite difference equations for the nodal values, treats the diffusion terms implicitly and is unconditionally stable if the reaction terms are treated implicitly. The method is very accurate for problems with the interior and/or boundary layers. For a system of two nonlinearly-coupled, one-dimensional reaction–diffusion equations, the formation, propagation and merging of reactive fronts have been found to be strong function of the diffusion coefficients and reaction rates. For asymmetric ignition, it has been found that, after front merging, the temperature and concentration profiles are almost independent of the ignition conditions.Originality/valueA new, conservative, transversal method of lines that treats the diffusion terms implicitly and provides piecewise exponential solutions in space without the need for interpolation is presented and applied to someone. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Numerical Methods for Heat and Fluid Flow Emerald Publishing

A conservative, piecewise–analytical, transversal method of lines for reaction–diffusion equations

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References (26)

Publisher
Emerald Publishing
Copyright
© Emerald Publishing Limited
ISSN
0961-5539
DOI
10.1108/hff-01-2019-0025
Publisher site
See Article on Publisher Site

Abstract

The purpose of this paper is to develop a new transversal method of lines for one-dimensional reaction–diffusion equations that is conservative and provides piecewise–analytical solutions in space, analyze its truncation errors and linear stability, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficients, reaction rate terms and initial conditions on wave propagation and merging.Design/methodology/approachA conservative, transversal method of lines based on the discretization of time and piecewise analytical integration of the resulting two-point boundary-value problems subject to the continuity of the dependent variables and their fluxes at the control-volume boundaries, is presented. The method provides three-point finite difference expressions for the nodal values and continuous solutions in space, and its accuracy has been determined first analytically and then assessed in numerical experiments of reaction-diffusion problems, which exhibit interior and/or boundary layers.FindingsThe transversal method of lines presented here results in three-point finite difference equations for the nodal values, treats the diffusion terms implicitly and is unconditionally stable if the reaction terms are treated implicitly. The method is very accurate for problems with the interior and/or boundary layers. For a system of two nonlinearly-coupled, one-dimensional reaction–diffusion equations, the formation, propagation and merging of reactive fronts have been found to be strong function of the diffusion coefficients and reaction rates. For asymmetric ignition, it has been found that, after front merging, the temperature and concentration profiles are almost independent of the ignition conditions.Originality/valueA new, conservative, transversal method of lines that treats the diffusion terms implicitly and provides piecewise exponential solutions in space without the need for interpolation is presented and applied to someone.

Journal

International Journal of Numerical Methods for Heat and Fluid FlowEmerald Publishing

Published: Oct 17, 2019

Keywords: Conservation; Finite; Transversal method of lines; Piecewise–analytical solution; Finite–volume technique; Nonlinear reaction–diffusion equations

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