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.
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
With the initial condition,
The boundary conditions arewhere is the diffusion coefficient, is the reactive factor, is the distance, is the time, and is the population density. , and 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 , whereas the nonlinear local multiplication or reaction is observed along [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 in h [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 domain in equation (1) be divided into equal subintervals with uniform mesh size which can be represented as , such that for . Similarly, the temporal solution domain can also be divided into equal subintervals with uniform width , such that for .
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 time level and the other is approximated at time level.
2.1. Fourth-Order Compact Scheme
From Taylor’s series expansion, the first and second derivatives of can be approximated as follows:
Here,
Substituting equation (5) into (1) yields
Differentiating equation (1) twice with respect to the space variable and rearranging, we obtain
Replacing the first and second derivatives of in equation (8) by equations (4) and (5), respectively, we get
Now, substituting equation (9) into (7) and rearranging yields
Using the method of lagging to linearize the nonlinear terms, the Crank–Nicolson discretization of equation (10) becomes
Using the expressions in equation (6) into (11) and simplifying, we get the following tridiagonal system of linear equations: .
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
Replacing by in equation (13) to avoid the use of and in complex number and simplifying, we arrive at
Replacing by and solving for , we get
Since , for every value of .
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 .
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 grids coarser and finer and also assume that the following difference equations for solving equation (1) are valid.where and are the remainders of order and C is a constant independent of and . To eliminate between the two difference equations, multiply equation (16) by and equation (17) by and then combine the resulting difference equations on the coarser gird to get better approximate solution using the following:
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 error at is approximated by
The and error norms at can also be approximated by
Example 1. Consider the equation (1) [28, 30] with , in . For this equation, the initial condition and the boundary condition are, respectively, given byThe exact solution for this equation is given by
Example 2. Consider the equation (1) [29] for with boundary conditionsFor this problem, the initial condition isThe exact solution is
3.2. Discussion
In Table 1, pointwise absolute errors of Example 1 for different values of the spatial step-size with time step-size and 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, and errors and the computational running time are displayed for two values of time step-size () for different values of when . As we increase the value of from 0.5 to 5, the errors obtained by and 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 error norm better approximates the exact solution than the 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 and errors, and the results show that our method better approximates the 1D Fisher’s equation in equation (1).
In Figures 1 and 2, the numerical solution is sketched against the exact solution for different values of with and . 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.
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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.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to express their heartfelt thanks to Jimma University for funding this project work.