# Compact Finite Difference Scheme Combined with Richardson Extrapolation for Fisher’s Equation

Authors: Hailu Muleta Chemeda; Ayana Deressa Negassa; Feyisa Edosa Merga
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this study, the fourth-order compact finite difference scheme combined with Richardson extrapolation for solving the 1D Fisher’s equation is presented. First, the derivative involving the space variable is discretized by the fourth-order compact finite difference method. Then, the nonlinear term is linearized by the lagging method, and the derivative involving the temporal variable is discretized by the Crank–Nicolson scheme. The method is found to be unconditionally stable and fourth-order accurate in the direction of the space variable and second-order accurate in the direction of the temporal variable. When combined with the Richardson extrapolation, the order of the method is improved from fourth to sixth-order accurate in the direction of the space variable. The numerical results displayed in figures and tables show that the proposed method is efficient, accurate, and a good candidate for solving the 1D Fisher’s equation.


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## Body

## 1. Introduction

Mathematical modeling of most physical systems leads to linear/nonlinear partial differential equations (PDEs) in various fields of science. PDEs have enormous applications compared to ordinary differential equations (ODEs) such as in dynamics, electricity, heat transfer, electromagnetic theory, quantum mechanics, and so on [1].The 1D Fisher’s equation is given by(1)∂u∂t=β∂2u∂x2+αu1−u,x∈a,b⊆−∞,∞,t≥0.With the initial condition,(2)ux,0=fx,a≤x≤b.The boundary conditions are(3)ua,t=g1t,ub,t=g2t,t≥0,where β>0 is the diffusion coefficient, α>0 is the reactive factor, x is the distance, t is the time, and ux,t is the population density. fx,g1t, and g2t are the smooth functions on the given domain.The 1D Fisher’s equation in equation (1) was first proposed by Fisher [2] as a model for the spatial and temporal propagation of a virile gene in an infinite medium [3]. It can also be considered as a model equation for the evolution of a neutron population in a nuclear reactor [4, 5]. Equation (1) also describes the rate of the advance of a new advantageous gene within a population of a constant density occupying a one-dimensional habitat [6]. In equation (1), the effect of the linear diffusion is observed along uxx, whereas the nonlinear local multiplication or reaction is observed along u1−u [5, 7, 8]. Some of the application areas of equation (1) include gene propagation [3, 5], tissue engineering [9], combustion [10], and neurophysiology [8].Due to its wider applications in the real world problems, many researchers have been developing both analytical and numerical methods for solving Fisher’s equation. Gazdag and Canosa [6] applied accurate space derivatives (ASD) which can be carried out efficiently by the use of the fast Fourier transformation algorithm to the numerical solution of equation (1). The Petrov–Galerkin finite element method [11], Sinc collocation method [12], and the wavelet Galerkin method [13] have been applied for solving equation (1). Bastani and Salkuyeh [14] studied the numerical solution of equation (1) using the compact difference method (FDM). Hasnain et al. in [5] also studied equation (1) numerically using the Crank–Nicolson scheme combined with the Richardson extrapolation technique.A widely and frequently used numerical technique for Fisher’s equation is the FDM. However, the usual FDM shows shortcomings in computational accuracy. The two widely known methods used to improve such shortcomings are the application of the compact finite difference method and Richardson extrapolation technique [15].Due to its high accuracy and stability property, the compact finite difference method attracts much attention from many scholars for finding approximate solutions of various kinds of equations [16], and it is much more accurate than the corresponding explicit scheme of the same order [17]. A high-order compact finite difference method was applied for systems of reaction-diffusion equations in [18], and the Helmholtz equation was approximated by a sixth-order compact finite difference (CFD6) method in [19]. Dennis et al. [20] proposed the fourth-order CFDS for convection-diffusion problems. Bastani and Salkuyeh [14] had combined a CFD6 scheme for second derivative in space and a third-order total variation diminishing the Runge–Kutta (TVD-RK3) scheme in time to approximate Fisher’s equation.Another way for improving the accuracy and rate of convergence of the FDM is through the application of Richardson’s extrapolation (RE) provided that their error term is expressible as a polynomial or power series inh [21, 22]. Furthermore, RE does not require any knowledge of the underlying methodology except the order of accuracy, which guarantees the minimal intervention to the existing computational tools [23]. Gordin [24] applied the RE method to improve a fourth-order CFDS to sixth-order in 1D parabolic equations and Schrödinger-type equations. Compared to low-order methods, high-order methods can achieve satisfactory errors on much coarser grids and thus greatly curtail the computational cost [25].Although mathematical properties of Fisher’s equation and plenty of discussion are available in the literature, majority of them do not address the import properties such as stability analysis, order of convergence, and consistence of the underlying numerical method. The aim of this study is to develop a higher-order numerical method for solving the 1D Fisher’s equation. We also establish the stability condition and order of convergence of the proposed method.


## 2. Mathematical Formulation and Analysis of the Proposed Method

Let the spatial solution domaina,b in equation (1) be divided into N equal subintervals with uniform mesh size h which can be represented as h=b−a/N, such that xi=a+ih for i=0,1,2,…,N. Similarly, the temporal solution domain 0,T can also be divided into M equal subintervals with uniform width k=T/M, such that tj=jk for j=0,1,2,…,M.To linearize the nonlinear term in equation (1), the method of lagging [26] is used in such a way one of the nonlinear term is approximated at j+1th time level and the other is approximated at jth time level.


### 2.1. Fourth-Order Compact Scheme

From Taylor’s series expansion, the first and second derivatives ofu can be approximated as follows:(4)ui′=δxui−h26ui3−h4120ui5+Oh6,(5)ui″=δx2ui−h212ui4−h4360ui6+Oh6.Here,(6)δxui=ui+1−ui−12h,δx2ui=ui+1−2ui+ui−1h2.Substituting equation (5) into (1) yields(7)∂ui∂t=βδx2ui−h212ui4+αui1−ui.Differentiating equation (1) twice with respect to the space variable and rearranging, we obtain(8)∂4ui∂x4=1β∂2∂x2∂ui∂t+2αβ∂ui∂x2−αβ1−2ui∂2ui∂x2.Replacing the first and second derivatives ofu in equation (8) by equations (4) and (5), respectively, we get(9)∂4ui∂x4=1β∂2∂x2∂ui∂t+2αβδxui2−αβ1−2uiδx2ui.Now, substituting equation (9) into (7) and rearranging yields(10)1+h212δx2∂ui∂t=βδx2ui−αh26δxui2−αh2121−2uiδx2ui+αui1−ui.Using the method of lagging to linearize the nonlinear terms, the Crank–Nicolson discretization of equation (10) becomes(11)1+h212δx2uij+1−uijk=βδx2uij+1+uij2−αh26δxuij+1δxuij−αh212δx2uij+1+uij2+αh26uij+1.δx2uij+αuij+11−uij.Using the expressions in equation (6) into (11) and simplifying, we get the following tridiagonal system of linear equations:(12)Aiui+1j+1+Biuij+1+Ciui−1j+1=Di,whereAi=2−12r+αk+αkui+1j−ui−1j,Bi=20+24r−26αk−4αkui+1j−8uij+ui−1j,Ci=2−12r+αk−αkui+1j−ui−1j,Di=2+12r−αkui+1j+ui−1j+20−24r+2αkuij,r=βk/h2.Equation (12) is called the fourth-order compact Crank–Nicolson scheme for solving the 1D Fisher’s equation.


### 2.2. Stability Analysis

One of the requirements of stability analysis by Von Neumann’s method is the linearity of the difference equations [27]. The linear form of equation (10) is given by(13)1+h212δx2∂ui∂t=βδx2ui−αh212δx2ui−αh212δx2ui1−constant+αui1−constant.Replacingih,jk by mh,nk in equation (13) to avoid the use of i and j in complex number and simplifying, we arrive at(14)1−6r+αkum+1n+1+10+12r+14αkumn+1+1−6r+αkum−1n+1=1+6r−αkum+1n+10−12r+2αkumn+1+6r−αkum−1n.Replacingumn by ξneimθ and solving for ξ, we get(15)ξ=1+6r−αkeiθ+e−iθ+10−12r+2αk1−6r+αkeiθ+e−iθ+10+12r+14αk=21+6r−αkcosθ+10−12r+2αk21−6r+αkcosθ+10+12r+14αk,ξ=21+6r−αkcosθ+10−12r+2αk21−6r+αkcosθ+10+12r+14αk≤21+6r−αkcosθ+10−12r+2αk21−6r+αkcosθ+10+12r+14αk.Since0≤cosθ≤1, ξ<1 for every value of r>0.Hence, the proposed method is unconditionally stable.


### 2.3. Order of the Proposed Method

To discretize the governing equation in equation (1), the first fourth-order compact finite difference method is applied in the space direction, and then, Crank–Nicolson is applied to discretize the derivative involving the temporal variable. Thus, the proposed method in equation (12) has fourth-order accuracy in the space direction and second-order accuracy in the temporal direction. In another word, the order of accuracy of the proposed method is Oh4+Ok2.


### 2.4. Application of Richardson Extrapolation

One of the advantages of the RE technique is that it increases the accuracy of the approximate solutions of the given differential equation [15, 28]. It can also be used to accelerate the convergence of the underlying method [22]. The aim is to apply the difference scheme on two consecutive grids and then combine the resulting solutions to obtain higher-order approximate solutions [21].Consider two consecutive gridsΩhk coarser and Ωh/2k finer and also assume that the following difference equations for solving equation (1) are valid.(16)Uhk=ux,t+Ch4+rh,k,xi,tj∈Ωhk,(17)Uh/2k=ux,t+Ch416+rh2,k,xi,tj∈Ωh/2k,where rh,k and rh/2,k are the remainders of order Oh6,k2 and C is a constant independent of h and k. To eliminate Ch4 between the two difference equations, multiply equation (16) by −1/15 and equation (17) by 16/15 and then combine the resulting difference equations on the coarser gird Ωhk to get better approximate solution using the following:(18)ux,t=16Uh/2k−Uhk15+Oh6,k2,xi,tj∈Ωhk.


## 2.1. Fourth-Order Compact Scheme

From Taylor’s series expansion, the first and second derivatives ofu can be approximated as follows:(4)ui′=δxui−h26ui3−h4120ui5+Oh6,(5)ui″=δx2ui−h212ui4−h4360ui6+Oh6.Here,(6)δxui=ui+1−ui−12h,δx2ui=ui+1−2ui+ui−1h2.Substituting equation (5) into (1) yields(7)∂ui∂t=βδx2ui−h212ui4+αui1−ui.Differentiating equation (1) twice with respect to the space variable and rearranging, we obtain(8)∂4ui∂x4=1β∂2∂x2∂ui∂t+2αβ∂ui∂x2−αβ1−2ui∂2ui∂x2.Replacing the first and second derivatives ofu in equation (8) by equations (4) and (5), respectively, we get(9)∂4ui∂x4=1β∂2∂x2∂ui∂t+2αβδxui2−αβ1−2uiδx2ui.Now, substituting equation (9) into (7) and rearranging yields(10)1+h212δx2∂ui∂t=βδx2ui−αh26δxui2−αh2121−2uiδx2ui+αui1−ui.Using the method of lagging to linearize the nonlinear terms, the Crank–Nicolson discretization of equation (10) becomes(11)1+h212δx2uij+1−uijk=βδx2uij+1+uij2−αh26δxuij+1δxuij−αh212δx2uij+1+uij2+αh26uij+1.δx2uij+αuij+11−uij.Using the expressions in equation (6) into (11) and simplifying, we get the following tridiagonal system of linear equations:(12)Aiui+1j+1+Biuij+1+Ciui−1j+1=Di,whereAi=2−12r+αk+αkui+1j−ui−1j,Bi=20+24r−26αk−4αkui+1j−8uij+ui−1j,Ci=2−12r+αk−αkui+1j−ui−1j,Di=2+12r−αkui+1j+ui−1j+20−24r+2αkuij,r=βk/h2.Equation (12) is called the fourth-order compact Crank–Nicolson scheme for solving the 1D Fisher’s equation.


## 2.2. Stability Analysis

One of the requirements of stability analysis by Von Neumann’s method is the linearity of the difference equations [27]. The linear form of equation (10) is given by(13)1+h212δx2∂ui∂t=βδx2ui−αh212δx2ui−αh212δx2ui1−constant+αui1−constant.Replacingih,jk by mh,nk in equation (13) to avoid the use of i and j in complex number and simplifying, we arrive at(14)1−6r+αkum+1n+1+10+12r+14αkumn+1+1−6r+αkum−1n+1=1+6r−αkum+1n+10−12r+2αkumn+1+6r−αkum−1n.Replacingumn by ξneimθ and solving for ξ, we get(15)ξ=1+6r−αkeiθ+e−iθ+10−12r+2αk1−6r+αkeiθ+e−iθ+10+12r+14αk=21+6r−αkcosθ+10−12r+2αk21−6r+αkcosθ+10+12r+14αk,ξ=21+6r−αkcosθ+10−12r+2αk21−6r+αkcosθ+10+12r+14αk≤21+6r−αkcosθ+10−12r+2αk21−6r+αkcosθ+10+12r+14αk.Since0≤cosθ≤1, ξ<1 for every value of r>0.Hence, the proposed method is unconditionally stable.


## 2.3. Order of the Proposed Method

To discretize the governing equation in equation (1), the first fourth-order compact finite difference method is applied in the space direction, and then, Crank–Nicolson is applied to discretize the derivative involving the temporal variable. Thus, the proposed method in equation (12) has fourth-order accuracy in the space direction and second-order accuracy in the temporal direction. In another word, the order of accuracy of the proposed method is Oh4+Ok2.


## 2.4. Application of Richardson Extrapolation

One of the advantages of the RE technique is that it increases the accuracy of the approximate solutions of the given differential equation [15, 28]. It can also be used to accelerate the convergence of the underlying method [22]. The aim is to apply the difference scheme on two consecutive grids and then combine the resulting solutions to obtain higher-order approximate solutions [21].Consider two consecutive gridsΩhk coarser and Ωh/2k finer and also assume that the following difference equations for solving equation (1) are valid.(16)Uhk=ux,t+Ch4+rh,k,xi,tj∈Ωhk,(17)Uh/2k=ux,t+Ch416+rh2,k,xi,tj∈Ωh/2k,where rh,k and rh/2,k are the remainders of order Oh6,k2 and C is a constant independent of h and k. To eliminate Ch4 between the two difference equations, multiply equation (16) by −1/15 and equation (17) by 16/15 and then combine the resulting difference equations on the coarser gird Ωhk to get better approximate solution using the following:(18)ux,t=16Uh/2k−Uhk15+Oh6,k2,xi,tj∈Ωhk.


## 3. Results, Discussion, and Conclusion

### 3.1. Numerical Test Examples

To validate the performance of the proposed method, we considered two numerical examples whose exact solution is available. The pointwise absolute errorε at t=T is approximated by(19)ε=uexactxi,T−uapproxxi,T,1≤i≤N−1.TheL∞ and L2 error norms at t=T can also be approximated by(20)L∞=max1≤i≤N−1uexactxi,T−uapproxxi,T,L2=1N∑i=1Nuexactxi,T−uapproxxi,T2.Example 1.
Consider the equation (1) [28, 30] with β=1, α=6 in 0,1. For this equation, the initial condition and the boundary condition are, respectively, given by(21)ux,0=1+ex−2;0≤x≤1,u0,t=1+e−5t−2,u1,t=1+e1−5t−2;0≤t≤T.
The exact solution for this equation is given by(22)ux,t=1+expx−5t−2;0≤x≤1,t≥0.Example 2.
Consider the equation (1) [29] for x∈−0.2,0.8 with boundary conditions(23)limx⟶−∞ux,t=1,limx⟶−∞ux,t=0.
For this problem, the initial condition is(24)ux,0=1+expα6x−2.
The exact solution is(25)ux,t=1+expα6x−α65t−2.


### 3.2. Discussion

In Table1, pointwise absolute errors of Example 1 for different values of the spatial step-size with time step-size Δt=0.001 and T=0.01 are tabulated. The displayed results and the comparison of the pointwise absolute errors obtained by the present method and the Keller Box method in [26], show that the present method approximates the exact solution very well. In Table 2, L2 and L∞ errors and the computational running time are displayed for two values of time step-size (Δt=0.0001and0.00005) for different values of T when h=0.025. As we increase the value of T from 0.5 to 5, the errors obtained by L2 and L∞ decrease rapidly; however, the computational time increases. The numerical results given in Table 3 clearly show that the present method is a good candidate for solving the 1D Fisher’s equation. Furthermore, it can be clearly seen from Tables 2 and 3 that the L2 error norm better approximates the exact solution than the L∞ error norm. In Table 4, the numerical results of the present method are compared against the results generated by Agbavon et al. in [30] in terms of L1 and L∞ errors, and the results show that our method better approximates the 1D Fisher’s equation in equation (1).Table 1
Comparison between pointwise errors of Example1 for Δt=0.001 and T=0.01 by the modified Keller Box method [26] and the present method.
MethodxN=10N=20N=40N=80Present method0.12.7048e − 057.8826e − 064.6663e − 064.7429e − 060.21.9913e − 062.2025e − 064.5380e − 066.2687e − 060.36.5613e − 066.7781e − 074.8015e − 066.9826e − 060.47.8298e − 068.4649e − 075.1902e − 067.3619e − 060.57.2517e − 061.3141e − 065.4721e − 067.5199e − 060.66.5162e − 061.6795e − 065.5846e − 067.4891e − 060.76.1483e − 061.8155e − 065.4960e − 067.2603e − 060.86.9525e − 061.4235e − 064.9992e − 066.6302e − 060.99.8358e − 061.3214e − 073.3749e − 064.7781e − 06Method in [26]0.13.7e − 052.9e − 052.7e − 052.6e − 050.24.6e − 053.5e − 053.3e − 053.2e − 050.34.4e − 053.5e − 052.6e − 052.6e − 050.43.6e − 052.8e − 052.6e − 052.6e − 050.53.7e − 053.1e − 052.9e − 052.8e − 050.63.4e − 052.8e − 052.6e − 052.5e − 050.73.1e − 052.5e − 052.4e − 052.3e − 050.82.6e − 052.1e − 052.0e − 051.9e − 050.91.8e − 051.5e − 051.4e − 051.3e − 05Table 2
Comparison betweenL2 and L∞ errors norms of Example 1 for Δt=0.0001 and 0.00005 when h=0.025 for different values of T.
TΔt=0.0001Δt=0.00005L2L∞CPU time (s)L2L∞CPU time (s)0.51.9230e − 052.7178e − 051.2256133.1680e − 054.4632e − 052.42200911.8592e − 062.4774e − 062.6522877.8558e − 071.3290e − 064.82444621.7754e − 082.4029e − 084.7324499.1469e − 091.2185e − 089.55494531.1986e − 101.6225e − 107.1471986.1787e − 118.2297e − 1114.69112148.2862e − 131.1298e − 129.5600273.4944e − 134.6230e − 1319.79349358.8105e − 151.6431e − 1412.0565411.7150e − 132.4192e − 1324.023544Table 3
Comparison betweenL2 and L∞ errors norms for Example 2 with Δt=0.0001 and 0.00005 when h=0.025 for different values of T.
TΔt=0.0001Δt=0.00005L2L∞CPU time (s)L2L∞CPU time (s)0.13.8809e − 166.6613e − 160.5976.5015e − 161.1102e − 151.0100.23.1924e − 166.6613e − 160.9876.4294e − 161.1102e − 151.8160.33.2715e − 166.6613e − 161.4186.5659e − 161.1102e − 152.8490.43.2715e − 166.6613e − 161.4596.5015e − 161.1102e − 153.6640.53.1113e − 166.6613e − 161.8086.4294e − 161.1102e − 154.513Table 4
L1, L2, and L∞ errors and computational time for Example 2 at time T=0.0025 with the space step-length h=0.01.
Present methodTime step (k)L1 errorL2 errorL∞ errorCPU time (s)T/523.5497e − 023.5232e − 031.5153e − 020.340T/1002.5420e − 021.7665e − 037.7685e − 030.443T/2001.7940e − 028.4546e − 043.8557e − 030.705T/3001.4662e − 025.4901e − 042.5648e − 030.958T/4001.2720e − 024.0708e − 041.9218e − 031.182T/5001.1400e − 023.2679e − 041.5368e − 031.516T/10008.5497e − 031.9788e − 047.6855e − 042.816T/12008.3671e − 031.8535e − 046.4073e − 043.435T/14008.2355e − 031.7903e − 045.4948e − 043.848T/16008.1384e − 031.7590e − 044.8106e − 044.425Method in [29]T/522.8968e − 01—1.52000.648T/1006.5144e − 01—6.9877e − 010.719T/2003.2980e − 02—3.9927e − 010.672T/3002.0727e − 02—2.5613e − 010.693T/4001.4262e − 02—1.7830e − 010.708T/5001.0268e − 02—1.2943e − 010.730T/10002.0059e − 03—2.6389e − 020.904T/12006.2983e − 03—9.0013e − 031.233T/14004.2509e − 04—4.3458e − 031.141T/16001.1913e − 03—1.3138e − 021.366In Figures1 and 2, the numerical solution is sketched against the exact solution for different values of T with Δt=0.0001 and h=0.025. From Figures 1 and 2, one can clearly see that the graphs of the approximate and the exact solutions overlap, showing that there is a good agreement between the approximate and exact solutions. Furthermore, the profiles of the approximate and numerical solutions shown in Figures 3 and 4 also reveal that the present method approximates the exact solution very well.Figure 1
Numerical and exact solutions for Example1 when h=0.025 and Δt=0.0001 for different values of T.Figure 2
Numerical and exact solutions for Example2 when h=0.01 and Δt=5e−06 for some different values of time.Figure 3
Solution profiles of numerical and exact solutions for Example1 when h=0.025 and Δt=0.0001 for different values of T.
(a)(b)Figure 4
Numerical and exact solution profiles for Example2 when h = 0.01 and Δt=5e−06 for some different values of time.
(a)(b)


### 3.3. Conclusion

In this study, the fourth-order compact Crank–Nicolson scheme combined with Richardson extrapolation for solving the 1D Fisher’s equation presented. First, the derivative envolving the space variable is discretized by the fourth-order compact finite difference method. Then, the nonlinear term in the governing equation is linearized by the lagging method and the derivative involving the temporal variable is discretized by the modified Crank–Nicolson scheme. The method is found to be unconditionally stable and fourth-order accurate in the direction of the space variable and second-order accurate in the direction of the temporal variable. When combined with the Richardson extrapolation, the order of the method is improved from fourth to sixth-order accurate in the direction of the space variable. To validate the applicability and efficiency of the proposed method, two numerical examples are solved, and the results are presented in tables and figures. From the results displayed in tables and figures, we can conclude that the the presnt method is efficient, accurate, and better approximates the 1D Fisher’s equation than some of the methods presented in the literature.


## 3.1. Numerical Test Examples

To validate the performance of the proposed method, we considered two numerical examples whose exact solution is available. The pointwise absolute errorε at t=T is approximated by(19)ε=uexactxi,T−uapproxxi,T,1≤i≤N−1.TheL∞ and L2 error norms at t=T can also be approximated by(20)L∞=max1≤i≤N−1uexactxi,T−uapproxxi,T,L2=1N∑i=1Nuexactxi,T−uapproxxi,T2.Example 1.
Consider the equation (1) [28, 30] with β=1, α=6 in 0,1. For this equation, the initial condition and the boundary condition are, respectively, given by(21)ux,0=1+ex−2;0≤x≤1,u0,t=1+e−5t−2,u1,t=1+e1−5t−2;0≤t≤T.
The exact solution for this equation is given by(22)ux,t=1+expx−5t−2;0≤x≤1,t≥0.Example 2.
Consider the equation (1) [29] for x∈−0.2,0.8 with boundary conditions(23)limx⟶−∞ux,t=1,limx⟶−∞ux,t=0.
For this problem, the initial condition is(24)ux,0=1+expα6x−2.
The exact solution is(25)ux,t=1+expα6x−α65t−2.


## 3.2. Discussion

In Table1, pointwise absolute errors of Example 1 for different values of the spatial step-size with time step-size Δt=0.001 and T=0.01 are tabulated. The displayed results and the comparison of the pointwise absolute errors obtained by the present method and the Keller Box method in [26], show that the present method approximates the exact solution very well. In Table 2, L2 and L∞ errors and the computational running time are displayed for two values of time step-size (Δt=0.0001and0.00005) for different values of T when h=0.025. As we increase the value of T from 0.5 to 5, the errors obtained by L2 and L∞ decrease rapidly; however, the computational time increases. The numerical results given in Table 3 clearly show that the present method is a good candidate for solving the 1D Fisher’s equation. Furthermore, it can be clearly seen from Tables 2 and 3 that the L2 error norm better approximates the exact solution than the L∞ error norm. In Table 4, the numerical results of the present method are compared against the results generated by Agbavon et al. in [30] in terms of L1 and L∞ errors, and the results show that our method better approximates the 1D Fisher’s equation in equation (1).Table 1
Comparison between pointwise errors of Example1 for Δt=0.001 and T=0.01 by the modified Keller Box method [26] and the present method.
MethodxN=10N=20N=40N=80Present method0.12.7048e − 057.8826e − 064.6663e − 064.7429e − 060.21.9913e − 062.2025e − 064.5380e − 066.2687e − 060.36.5613e − 066.7781e − 074.8015e − 066.9826e − 060.47.8298e − 068.4649e − 075.1902e − 067.3619e − 060.57.2517e − 061.3141e − 065.4721e − 067.5199e − 060.66.5162e − 061.6795e − 065.5846e − 067.4891e − 060.76.1483e − 061.8155e − 065.4960e − 067.2603e − 060.86.9525e − 061.4235e − 064.9992e − 066.6302e − 060.99.8358e − 061.3214e − 073.3749e − 064.7781e − 06Method in [26]0.13.7e − 052.9e − 052.7e − 052.6e − 050.24.6e − 053.5e − 053.3e − 053.2e − 050.34.4e − 053.5e − 052.6e − 052.6e − 050.43.6e − 052.8e − 052.6e − 052.6e − 050.53.7e − 053.1e − 052.9e − 052.8e − 050.63.4e − 052.8e − 052.6e − 052.5e − 050.73.1e − 052.5e − 052.4e − 052.3e − 050.82.6e − 052.1e − 052.0e − 051.9e − 050.91.8e − 051.5e − 051.4e − 051.3e − 05Table 2
Comparison betweenL2 and L∞ errors norms of Example 1 for Δt=0.0001 and 0.00005 when h=0.025 for different values of T.
TΔt=0.0001Δt=0.00005L2L∞CPU time (s)L2L∞CPU time (s)0.51.9230e − 052.7178e − 051.2256133.1680e − 054.4632e − 052.42200911.8592e − 062.4774e − 062.6522877.8558e − 071.3290e − 064.82444621.7754e − 082.4029e − 084.7324499.1469e − 091.2185e − 089.55494531.1986e − 101.6225e − 107.1471986.1787e − 118.2297e − 1114.69112148.2862e − 131.1298e − 129.5600273.4944e − 134.6230e − 1319.79349358.8105e − 151.6431e − 1412.0565411.7150e − 132.4192e − 1324.023544Table 3
Comparison betweenL2 and L∞ errors norms for Example 2 with Δt=0.0001 and 0.00005 when h=0.025 for different values of T.
TΔt=0.0001Δt=0.00005L2L∞CPU time (s)L2L∞CPU time (s)0.13.8809e − 166.6613e − 160.5976.5015e − 161.1102e − 151.0100.23.1924e − 166.6613e − 160.9876.4294e − 161.1102e − 151.8160.33.2715e − 166.6613e − 161.4186.5659e − 161.1102e − 152.8490.43.2715e − 166.6613e − 161.4596.5015e − 161.1102e − 153.6640.53.1113e − 166.6613e − 161.8086.4294e − 161.1102e − 154.513Table 4
L1, L2, and L∞ errors and computational time for Example 2 at time T=0.0025 with the space step-length h=0.01.
Present methodTime step (k)L1 errorL2 errorL∞ errorCPU time (s)T/523.5497e − 023.5232e − 031.5153e − 020.340T/1002.5420e − 021.7665e − 037.7685e − 030.443T/2001.7940e − 028.4546e − 043.8557e − 030.705T/3001.4662e − 025.4901e − 042.5648e − 030.958T/4001.2720e − 024.0708e − 041.9218e − 031.182T/5001.1400e − 023.2679e − 041.5368e − 031.516T/10008.5497e − 031.9788e − 047.6855e − 042.816T/12008.3671e − 031.8535e − 046.4073e − 043.435T/14008.2355e − 031.7903e − 045.4948e − 043.848T/16008.1384e − 031.7590e − 044.8106e − 044.425Method in [29]T/522.8968e − 01—1.52000.648T/1006.5144e − 01—6.9877e − 010.719T/2003.2980e − 02—3.9927e − 010.672T/3002.0727e − 02—2.5613e − 010.693T/4001.4262e − 02—1.7830e − 010.708T/5001.0268e − 02—1.2943e − 010.730T/10002.0059e − 03—2.6389e − 020.904T/12006.2983e − 03—9.0013e − 031.233T/14004.2509e − 04—4.3458e − 031.141T/16001.1913e − 03—1.3138e − 021.366In Figures1 and 2, the numerical solution is sketched against the exact solution for different values of T with Δt=0.0001 and h=0.025. From Figures 1 and 2, one can clearly see that the graphs of the approximate and the exact solutions overlap, showing that there is a good agreement between the approximate and exact solutions. Furthermore, the profiles of the approximate and numerical solutions shown in Figures 3 and 4 also reveal that the present method approximates the exact solution very well.Figure 1
Numerical and exact solutions for Example1 when h=0.025 and Δt=0.0001 for different values of T.Figure 2
Numerical and exact solutions for Example2 when h=0.01 and Δt=5e−06 for some different values of time.Figure 3
Solution profiles of numerical and exact solutions for Example1 when h=0.025 and Δt=0.0001 for different values of T.
(a)(b)Figure 4
Numerical and exact solution profiles for Example2 when h = 0.01 and Δt=5e−06 for some different values of time.
(a)(b)


## 3.3. Conclusion

In this study, the fourth-order compact Crank–Nicolson scheme combined with Richardson extrapolation for solving the 1D Fisher’s equation presented. First, the derivative envolving the space variable is discretized by the fourth-order compact finite difference method. Then, the nonlinear term in the governing equation is linearized by the lagging method and the derivative involving the temporal variable is discretized by the modified Crank–Nicolson scheme. The method is found to be unconditionally stable and fourth-order accurate in the direction of the space variable and second-order accurate in the direction of the temporal variable. When combined with the Richardson extrapolation, the order of the method is improved from fourth to sixth-order accurate in the direction of the space variable. To validate the applicability and efficiency of the proposed method, two numerical examples are solved, and the results are presented in tables and figures. From the results displayed in tables and figures, we can conclude that the the presnt method is efficient, accurate, and better approximates the 1D Fisher’s equation than some of the methods presented in the literature.



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