An Improved Error Estimate for a Numerical Method for a System of Coupled Singularly Perturbed Reaction-diffusion Equations

AbstractWe consider a central difference scheme for the numerical solution of a system of coupled reaction-diffusion equations. We show that the scheme is almost second-order convergent, uniformly in the perturbation parameter. We present the results of numerical experiments to confirm our theoretical results.

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CONSERVATIVE NUMERICAL METHOD FOR A SYSTEM OF SEMILINEAR SINGULARLY PERTURBED PARABOLIC REACTION‐DIFFUSION EQUATIONS

Mathematical Modelling and Analysis ◽

10.3846/1392-6292.2009.14.211-228 ◽

2009 ◽

Vol 14 (2) ◽

pp. 211-228 ◽

Cited By ~ 7

Author(s):

Lidia Shishkina ◽

Grigorii Shishkin

Keyword(s):

Unit Length ◽

Reaction Diffusion ◽

Perturbation Parameter ◽

Reaction Diffusion Equations ◽

Singularly Perturbed ◽

Diffusion Equations ◽

Finite Difference Schemes ◽

Vertical Strip ◽

Parabolic Boundary ◽

Conservative Numerical Method

On a vertical strip, a Dirichlet problem is considered for a system of two semilinear singularly perturbed parabolic reaction‐diffusion equations connected only by terms that do not involve derivatives. The highest‐order derivatives in the equations, having divergent form, are multiplied by the perturbation parameter e ; ϵ ∈ (0,1]. When ϵ → 0, the parabolic boundary layer appears in a neighbourhood of the strip boundary. Using the integro‐interpolational method, conservative nonlinear and linearized finite difference schemes are constructed on piecewise‐uniform meshes in the x 1 ‐axis (orthogonal to the boundary) whose solutions converge ϵ‐uniformly at the rate O (N1−2 ln2 N 1 + N 2 −2 + N 0 −1). Here N 1 + 1 and N 0 + 1 denote the number of nodes on the x 1‐axis and t‐axis, respectively, and N 2 + 1 is the number of nodes in the x 2‐axis on per unit length.

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Stability of Exact and Discrete Energy for Non-Fickian Reaction-Diffusion Equations with a Variable Delay

Abstract and Applied Analysis ◽

10.1155/2014/840573 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9 ◽

Cited By ~ 2

Author(s):

Dongfang Li ◽

Chao Tong ◽

Jinming Wen

Keyword(s):

Numerical Experiments ◽

Reaction Diffusion ◽

Reaction Diffusion Equations ◽

Diffusion Equations ◽

Variable Delay ◽

Discrete Energy ◽

Continuous Problems ◽

Rate Of Decay ◽

The Stability ◽

Theoretical Results

This paper is concerned with the stability of non-Fickian reaction-diffusion equations with a variable delay. It is shown that the perturbation of the energy function of the continuous problems decays exponentially, which provides a more accurate and convenient way to express the rate of decay of energy. Then, we prove that the proposed numerical methods are sufficient to preserve energy stability of the continuous problems. We end the paper with some numerical experiments on a biological model to confirm the theoretical results.

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ROBUST NUMERICAL METHOD FOR A SYSTEM OF SINGULARLY PERTURBED PARABOLIC REACTION‐DIFFUSION EQUATIONS ON A RECTANGLE

Mathematical Modelling and Analysis ◽

10.3846/1392-6292.2008.13.251-261 ◽

2008 ◽

Vol 13 (2) ◽

pp. 251-261 ◽

Cited By ~ 8

Author(s):

Lida Shishkina ◽

Grigory Shishkin

Keyword(s):

Reaction Diffusion ◽

Perturbation Parameter ◽

Reaction Diffusion Equations ◽

Singularly Perturbed ◽

Diffusion Equations ◽

Solution Technique ◽

Parabolic Boundary ◽

Finite Difference Approximations ◽

Parabolic Boundary Layer ◽

Mesh Technique

A Dirichlet problem is considered for a system of two singularly perturbed parabolic reaction‐diffusion equations on a rectangle. The parabolic boundary layer appears in the solution of the problem as the perturbation parameter ϵ tends to zero. On the basis of the decomposition solution technique, estimates for the solution and derivatives are obtained. Using the condensing mesh technique and the classical finite difference approximations of the boundary value problem under consideration, a difference scheme is constructed that converges ϵ‐uniformly at the rate O ‘N−2 ln2 N + N0 −1) , where N = mins Ns, s = 1, 2, Ns + 1 and N0 + 1 are the numbers of mesh points on the axis xs and on the axis t, respectively.

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