A fourth-order accurate, Numerov-type, three-point finite-difference discretization of electrochemical reaction-diffusion equations on nonuniform (exponentially expanding) spatial grids in one-dimensional space geometry

2004 ◽  
Vol 25 (12) ◽  
pp. 1515-1521 ◽  
Author(s):  
Les?aw K. Bieniasz
Keyword(s):  
Finite Difference ◽  
Electrochemical Reaction ◽  
Dimensional Space ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
One Dimensional ◽  
Finite Difference Discretization ◽  
Spatial Grids ◽  
Difference Discretization

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

2019 ◽  
Vol 29 (11) ◽  
pp. 4093-4129 ◽  
Author(s):  
J.I. Ramos
Keyword(s):  
Finite Difference ◽  
Boundary Layers ◽  
Diffusion Coefficients ◽  
Method Of Lines ◽  
Reaction Diffusion ◽  
Reaction Rates ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Content Type ◽  
One Dimensional

Purpose 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/approach A 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. Findings The 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/value A 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.

Numerical simulations of reaction–diffusion equations modeling prey–predator interaction with delay

2018 ◽  
Vol 11 (04) ◽  
pp. 1850054 ◽  
Author(s):  
Ishtiaq Ali ◽  
Ghulam Rasool ◽  
Saleh Alrashed
Keyword(s):  
Dimensional Space ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
One Dimensional ◽  
Natural Systems ◽  
Nonlinear Reaction ◽  
The Rich ◽  
Delay Differential ◽  
Type Ii Functional Response

To model biological systems one often uses ordinary and partial differential equations. These equations can be quite good at approximating observed behavior, but they suffer from the downfall of containing many parameters, often signifying quantities which cannot be determined experimentally. For the better understanding of complicated phenomena, the delay differential equation approach to model such phenomena is becoming more and more essential to capture the rich variety of dynamics observed in natural systems. In this study, we investigated numerically the influence of delay on the dynamics of nonlinear reaction–diffusion equations modeling prey–predator interaction using finite difference scheme subject to appropriate initial and boundary conditions. We first consider the prey–predator with Holling type II functional response where the growth of prey is assumed to be logistic in the sense of predator in one-dimensional space. The effect of delay was investigated with the help of simulations and is compared with the model equation without delay. The proposed method is then extended to two-dimensional space.

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