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American Journal of Computational Mathematics, 2018, 8, 55-67

http://www.scirp.org/journal/ajcm

ISSN Online: 2161-1211

ISSN Print: 2161-1203

DOI:

10.4236/ajcm.2018.81005 Mar. 16, 2018 55

American Journal of Computational Mathematics

Solving Stiff Reaction-Diffusion Equations Using

Exponential Time Differences Methods

H. A. Ashi

The Mathematical Department, King AbdulAziz University, Jeddah, KSA

Abstract

Reaction-diffusion equations modeling Predator-Prey interaction are of cu

rent interest. Standard approaches such as first-order (in time) finite diffe

ence schemes for approximating the solution are widely spread. Though,

this

paper shows that recent advance methods can be more favored. In this work

we have incorporated, throughout numerical comparison experiments, spe

tral methods, for the space discretization,

in conjunction with second and

fourth-

order time integrating methods for approximating the solution of the

reaction-

diffusion differential equations. The results have revealed that these

methods have advantages over the conventional methods, some of which

mention are: the ease of implementation, accuracy and CPU time.

Keywords

Finite Difference Methods, Exponential Integrator, Exponential Time

Differencing Method, Reaction-Diffusion System

1. Introduction

Numerical methods are important tools in investigating the solution’s behavior

of non-linear realistic models in biology [1] where no closed form solutions exist.

A class of these models is reaction diffusion (RD) problems that, for instant,

reproduce some of the complex pattern observed on the skin of certain animals.

This includes the development of coat patterns on mammals and the patterning

of butterfly wings [1].

Reaction diffusion models have been studied extensively since the RD theory

first proposed by

Turing [2] to describe the range of spatial patterns observed in

the developing embryo.

In recent years, several theoretical models, regarding spatial pattern, have

How to cite this paper:

Ashi, H.A. (2018

)

Solving Stiff Reaction

Diffusion Equations

Using Exponential Time Differences M

thods

American Journal of Computational

Mathematics

, 55-67.

https://doi.org/10.4236/ajcm.2018.81005

Received:

December 28, 2017

Accepted:

March 13, 2018

Published:

March 16, 2018

8 by author and

Scientific

Research Publishing Inc.

This work is licensed under the Creative

Commons Attribution

International

License (CC BY

4.0).

http://creativecommons.org/licenses/by/4.0/

Open Access

H. A. Ashi

DOI:

10.4236/ajcm.2018.81005 56 American

Journal of Computational Mathematics

been introduced and led to a better understanding of how these patterns arise

from certain mechanism. A general theoretical framework for studying pattern

formation in biological systems within a growing domain was developed in [3].

In addition, reaction diffusion equations modeling predator-prey interactions

have been paid attention and studied extensively [4] [5] [6]. In these models, the

most important element is the “ functional response”, the function that describes

the number of prey consumed by predator per unit time.

For the numerical solution of nonlinear reaction diffusion problems (most of

which are stiff

), fully explicit temporal methods are avoided due to the sever

restriction on the time step size imposed by the stiff diffusion term. Fully

implicit temporal schemes are not recommended since they have the task of

solving large implicit systems at each time step. Alternatively, several methods

were introduced. The vast majority of scientific community uses, for the spatial

discretization, lower-order centred finite differences schemes due to the ease of

implementation and incorporation of the boundary conditions in a straightforward

way. The time integration is carried out, usually, by using some types of low-order

time step methods such as, second-order fully implicit

Crank-Nicholson

methods.

The

Exponential Time Differencing (ETD) methods [12] in conjunction

with spectral methods [13] [14] [15] have emerged as viable alternatives to

classical schemes for a wide variety of problems. Their applications to solve

reaction-diffusion problems [6] [16] [17] [18] [19] have been steadily growing.

Spectral methods are popular for generating spectral accurate spatial derivatives,

through built-in codes in

Matlab [20]. ETD schemes recover the exact solution

to the linear part (which is generally the most stiff part of the system), and

integrate exactly an approximation of the non-linear terms. We may approximate

the non-linear parts by some polynomial in time that may be calculated using

previous steps of the integration process (producing multi-step ETD methods)

or by Runge-Kutta-like stages (resulting in ETD schemes of RK type), see [21]

for a comprehensive review. The numerical computation of the ETD coefficients,

which are functions related to the matrix exponential, arises some difficulties [22]

[23]. Fortunately, several numerical algorithms [19] [22]-[28] were introduced

to enhance the accuracy in computing these coefficients.

The work presented in this article is motivated by the theoretical and

computational study by

Gavie [5] for the dynamical properties of the 2-component

reaction-diffusion system modeling predator-prey interactions with the Holling

type

II functional response. The system displays a wide spectrum of ecologically

relevant behavior, including chaos. The model and an illustration of the

computational study conducted by

Garvie [5], using first-order finite difference

Stiff differential equations are

categorized as those whose solutions (or different components of a

single solution) evolve on very different time scales occurring simultaneously,

i.e.

the rates of change

of the various components of the solutions differ markedly. For a comprehensive revi

ew of this

phenomena see [7] [8] [9] [10] [11].

H. A. Ashi

DOI:

10.4236/ajcm.2018.81005 57

American Journal of Computational Mathematics

schemes, are briefly described in §2. In §3, we numerically approximate the

solution of the predator-prey model in two dimensions employing spectral

methods, for the space discretization, in conjunction with high-order time

integrating methods. The results of the numerical experiment are compared to

Garvie’s findings. Section §4 recaps our results and points out some

conclusions.

2. Model Problems

In this section we briefly give the 2-component reaction-diffusion non-dimensional

system, modeling predator-prey interactions with the Holling type

II functional

response and logistic growth of the prey, as follows (taken from [5]):

( ) ( )

( )

u u u vh au

v bvh au cv

∂

=∆+ − −

∂

=∆+ −

∂

(1)

The population densities of the prey and predators at time

and vector

position

are denoted by

( )

,utx

and

( )

,vtx

respectively.

∆

is the usual

Laplace operator in

3d ≤

space dimensions and the parameters

abc

and

are strictly positive. The domain (habitat), defined by

Ω

, is bounded and

configured with appropriate initial condition and zero-flux boundary conditions,

ensuring that no individual species can leave the domain. The “functional

response”

( )

h ⋅

represents the prey consumption rate per predator as a fraction

of the maximal consumption rate. It is assumed to be a

function satisfying

the following conditions:

•

(

)

h =

•

( )

lim 1

→∞

•

( )

h ⋅

is strictly increasing on

[

)

0,∞

A specific type II functional response with positive parameters

αβ

and

is given as follows [5]

(

) (

)

, , with 1 , , .

h au a b c

η η α βγ

= = = = =

Thus, the type of kinetics governed in this paper is

( ) ( )

( )

, 1 , , .

uv uv

f uv u u g uv v

αα

= −− = −

(2)

We consider the natural biological meaningful region

0, 0uv≥≥

with

bounded positive initial data, where the solution remains positive at all time.

From an environmental point of view, the choice of parameters for the

numerical simulation of the full reaction-diffusion system insure that both the

spatial and temporal local dynamics (the densities) of the predator and preys are

oscillatory. It is important to note that the above model takes into account the

invasion of the prey species by predators but does not include stochastic effects

or any influences from the environment.

H. A. Ashi

DOI:

10.4236/ajcm.2018.81005 58 American

Journal of Computational Mathematics

Computational Issues

For the two dimensional approximation, let

1, ,nN= 

and

, 0, ,

ij J=



where

and

denote the number of uniform subdivision of the time interval

[ ]

0,T

and the square domain

[ ] [ ]

, , , 0AB AB A BΩ= × <

respectively. The

forward difference in time and the five point central difference approximation of

the Laplace in two dimensions are defined as follows:

(

)

( )

, , 1 , 1 1, 1, ,

n nn

h ij ij ij i j i j ij

U UU t

UUUUU Uh

−

+−+−

∂= − ∆

∆= +++−

where

t TN∆=

and

( )

h BAJ= −

represent the time and space step

respectively. In addition, suppose that

and

denote the two dimensional

approximation to the solutions

and

respectively at the point

( )

i jn

xyt

where the time levels

t nt= ∆

Garvie [5] presents two first-order semi-implicit (in time) finite difference

schemes as follows:

, , , ,,

,, ,

ij ij

n n n nn

n ij h ij ij ij ij

ij ij

nn n

n ij h ij ij

U U U UU

VV V

δγ

−

∂ =∆ +− −

∂=∆+ −

(3)

1 11

, , , ,,

,, ,

ij ij

n n n nn

n ij h ij ij ij ij

ij ij

nn n

n ij h ij ij

U U U UU

VV V

δγ

−−

− −−

−

−−

−

∂=∆+− −

∂=∆+ −

(4)

with the given initial approximations

( ) ( )

,0 ,0

,, ,,

ij i j ij i j

U u xy V v xy= =

and zero-flux boundary conditions, for studying the dynamics of spatially

extended predator-prey interaction with logistic growth of the prey (1) in two

space dimensions. The “analytical solution” of the predator-prey system (1) is

approximated using schemes (3) and (4) on a fine mesh and small time step. The

linear system of algebraic equations, resulting from the solution of the problem

using either schemes, is sparse, banded and has a simple structure (tridiagonal and

block tridiagonal system in one and two dimensions respectively). In addition, the

coefficient matrices of the linear system are strictly diagonally dominant, which

guarantee the convergence of the

GMRES algorithm (an iterative solver) used to

solve the system (in two dimensions), see [5].

The convergence of Schemes (3) and (4) was decided on upon reducing the

time step until the difference between the approximations from either schemes is

negligible. On the other hand, the space step is kept sufficiently small to see

clearly the qualitative feature of the solutions.

The numerical experiments conducted by

Garvie had numerical and ecological

implications. Numerically,

Garvie showed that

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