Accelerated monotone iterative methods for finite difference equations of reaction-diffusion

Abstract

This paper is concerned with numerical methods for a finite difference system of reaction-diffusion-convection equation under nonlinear boundary condition. Various monotone iterative methods are presented, and each of these methods leads to an existence-comparison theorem as well as a computational algorithm for numerical solutions. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration, and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function. Application is given to a model problem from chemical engineering, and some numerical results, including a test problem with known analytical solution, are presented to illustrate the various rates of convergence of the iterations.

Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator–Prey Interactions in MATLAB

“…The finite-difference method (Hildebrand, 1968;Morton and Mayers, 1996;Richtmyer and Morton, 1967) is widely used by the scientific community for the numerical solution of reaction-diffusion equations; however, there are comparatively few studies that give stability and convergence results (see for example Ascher et al, 1995;Beckett and Mackenzie, 2001;Hoff, 1978;Jerome, 1984;Li et al, 1994;Mickens, 2003;Pao, 1998Pao, , 1999Pao, , 2002Pujol and Grimalt, 2002). The need for a rigorous numerical analysis is due to the well-known (but often neglected) fact that the dynamics of the approximations of nonlinear differential equations (DEs) can differ significantly from that of the original DEs themselves.…”

Finite difference reaction-diffusion equations with nonlinear diffusion coefficients

“…We shall show that by some appropriate choice of b i,n } converge monotonically from above and below, respectively, to a unique solution of (2.4). Although this approach is similar to that in [7]- [11] for semilinear parabolic equations where K(u) ≡ K is a constant,the…”

“…Clearly, n 0 > 0, and by the same argument as that in [9]- [11], n 0 ∈ S p . This ensures that (i 0 , n 0 ) ∈ Λ p .…”

Uniform convergent monotone iterates for semilinear singularly perturbed parabolic problems

“…An advantage of this accelerated approach is that it leads to sequences which converge either quadratically or nearly quadratically. In [8], an accelerated monotone iterative method for solving discrete parabolic boundary value problems is presented. In the recent paper [9], a combination of the accelerated monotone iterative method from [8] with monotone Picard iterates is constructed.…”