A Computational Analysis to Burgers Huxley Equation

: The efficiency of solving computationally partial differential equations can be profoundly highlighted by the creation of precise, higher-order compact numerical scheme that results in truly outstanding accuracy at a given cost. The objective of this article is to develop a highly accurate novel algorithm for two dimensional non-linear Burgers Huxley (BH) equations. The proposed compact numerical scheme is found to be free of superiors approximate oscillations across discontinuities, and in a smooth flow region, it efficiently obtained a high-order accuracy. In particular, two classes of higher-order compact finite difference schemes are taken into account and compared based on their computational economy. The stability and accuracy show that the schemes are unconditionally stable and accurate up to a two-order in time and to six-order in space. Moreover, algorithms and data tables illustrate the scheme efficiency and decisiveness for solving such non-linear coupled system. Efficiency is scaled in terms of L 2 and L ∞ norms, which validate the approxi-mated results with the corresponding analytical solution. The investigation of the stability requirements of the implicit method applied in the algorithm was carried out. Reasonable agreement was constructed under indistinguishable computational conditions. The proposed methods can be implemented for real-world problems, originating in engineering and science.


Introduction
This paper describes the multiplex schemes solution for two dimensional non-linear Burgers Huxley equation. Such an equation serves as the coupling between the Z xx , Z yy diffusive terms and Z(Z x + Z y ) the convectional phenomena. This equation is of high importance for showing a prototype model describing the interaction between reaction mechanisms, convection effects and diffusion transports. It is the combination of both Burgers & Huxley phenomena with non-linear term means reactions kind of characteristics behaviour, to capture some features of uid turbulence which caused by the effects of convection & diffusion [1][2][3]. It is a quantitative paradigm which deals with the ow of electric current through the surface membrane of a giant nerve bre. Nerve pulse propagation in nerve bres and wall motion in liquid crystals. Recently research has been measured to investigate two dimensional Burgers Huxley phenomena for understanding the various physical ows in uid theory [4][5][6] which leads to implementing a novel methodology for studying new insights [7,8]. It is worth mentioning that there is a vast amount of different approaches available in the literature to calculate the solutions of non-linear systems of partial differential equations. Seeking the Burgers Huxley equations numerical solution, wavelet collocation methods for the solution of Burgers Huxley equations [9] have already been studied in combination with variational iteration technique [10,11]. Moreover, the propagation of genes (Burger & Fisher) and Reaction-Diffusion (Gray Scott) models [12,13] investigated largely by the technique of computation [14]. On the other hand, optimal homotopy asymptotic & homotopy perturbation method was carried out to nd the approximate solution of Boussinesq-Burgers equations [15]. Finally, some novel techniques also take into account like chaos theory [16], nonlinear optics and fermentation process [17,18]. Wazwaz obtained the solitary wave solutions of one dimensional Burgers Huxley equation using tanh-coth method [19]. Hashim et al. [20,21] using Adomian Decomposition Method. Molabahrami et al. [22] used the homotopy analysis method to nd the solution of one dimensional Burger Huxley equation also E mova et al. [23] nd the travelling wave solution of such equation. Batiha et al. [24] used Hope-Cole transformation with Gao et al. [25] nd the exact solution of the generalized Burgers equation.
This research aims to deal with higher-order compact schemes with the nite difference methodology [8]. Our primary focus is to attain a compatible scheme which is highly ef cient and easy to implement with better accuracy. Although, Burgers Huxley equation can be in three dimensions still some features kept unexplored in the two-dimensional scenario. Let us explorer some new insights in BH equation which consists of the two-dimensional domain which can be written as: where Z = Z(l, m, t) is the unknown velocity & (l, m, t) ∈ × (0 T]. Laplacian can be de ned as with two dimensional behavior, is a non-linear reaction term. The coef cient ξ , η are advection and reactions coef cients accordingly with 0 < β < 1 & µ > 0. These parameters describe the interaction between reaction mechanisms, convection effects & diffusion transports [26,27]. Let us consider the initial condition, which can be seen from the upcoming Eq. (12). The Dirichlet boundary conditions are given by, where is a rectangular domain in R 2 & Z 0 , p 1 , p 2 , q 1 , q 2 are given suf ciently smooth functions, and Z(l, m, t) may represent unknown velocity, whereas Z.Z l , Z.Z m represents convection terms along with linear diffusion Z ll , Z mm . Such phenomena perpetuate the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon [28,29].
More generally, it is a challenging task for determining and preservation of physical properties like accuracy, stability, convergence criteria and design ef ciency for the given two-dimensional problem. This equation can be an effective procedure for the solution of various deterministic problems in physics, biology and chemical reactions. Also, deals in the investigation of the growth of colonies of bacteria consider population densities or sizes, which are non-negative variables. Most non-linear models of real-life problems are still very challenging to solve either numerically or theoretically. There has recently been much attention devoted to the search for better and more ef cient solution methods for determining a solution, analytical or numerical, to non-linear models [30,31]. In [31][32][33][34] authors present a method used to solve partial equations with the use of arti cial neural networks and an adaptive strategy to collocate them. To get the approximate solution of the partial differential equations Deep Neural Networks (DNNs) has been used, which shows impressive results in areas such as visual recognition [35]. Recently in [36], authors develop a numerical method with third-order temporal accuracy to solve time-dependent parabolic and rst-order hyperbolic partial differential equations. We focused on elaborating further by comparing analytical and numerical techniques.

Tanh-Coth Method
The dynamical balance between the non-linear reaction term and diffusive effects which constitute stable waveform after colliding with each other. In (1) the negative coef cients of Z ll , Z mm and Z 3 follow the physical behaviour of two dimensional BH Eq. (1). Such an equation can be converted into the non-linear ordinary differential equation which is as follows: Let σ = x − et, the wave variable which balances the non-linear reaction term (P µ,λ (Z)) where µ, λ are index values and diffusion transport (the highest derivative involved), we have M + 2 = 3, MM = 1. This enables us to set: Put M = 1 in (6) we get Let Y = tanh(γ σ ), and σ = ((x + y) − et) Then, Substitutes aforementioned in Eq. (6), we have the following solution to (7) 1 Arranging the coef cients of Y i , i ≥ 0, and equating these coef cients to zero, the system of algebraic equations in a 0 , a 1 , b 1 , γ and e are obtained. By solving the following set of the algebraic system of equations, we have the following form: In Eq. (10), the solution is of the form: Case 2: We found that a 1 = 0, From Cases 1 and 2, the kink solution is of the form: Now by solving (1) using the tanh-coth method, the analytical solution (kink solution ) is in a compact form in both cases is as follows: with initial condition: where Z is the unknown velocity, and γ & σ are wavenumbers which are developed during the solution of BH equation.

Description of Compact Schemes
Let us discretize the spatial domain which consists of N and M positive integers, such that h l and h m present step sizes along with l and m directions, respectively [37]. The spatial nodes can be denoted by l i , m j , namely, For the temporal domain, let us take dt as time-step discretization, τ = T/dt, with t n = Nτ . Also . . , w dt ) T , for any w ∈ ZZ τ , with some more notations: for n = 0, 1, 2, . . . , dt − 1.

Implementation Procedure:
Let us we divide (1) into two parts such as: Now considering one-dimensional steady convection-diffusion equation in the following form: where α 1 , α 11 are the constants while β 1 , β 11 are the convective velocities. F is the smooth functions of l and m may represent the reaction, vorticity. Now the three-point scheme is as follows: Now applying the Taylor series expansion to Eq. (14) we have the following results: where 0 ≤ n ≤ dt − 1 and the truncation error is By adding Eqs. (17) and (18) we have the following form (1) which yields: where Residual Apply Crank-Nicholson time discretization, which leads to: represents the truncation error [36][37][38][39]. The existence and uniqueness of the solutions of the scheme (21) can be easily found by positive de nite property. By applying operators and simplifying the (21) in compact form, The scheme is a system of linear equations based on variable Z n i, j , then after applying operators 21 can be written in the following way: where T 1 , T 11 , T 2 , T 22 , ff 1 , . . . , are all constants coef cients of Z n+1 i, j , Z n i, j and ff which includes α 1 , α 11 , β 1 , β 11 , h l , h m , τ and constant values.
The matrix form of the compact scheme is as follows By calculating and simplifying the terms, we have the following tridiagonal matrix if of the form: .
where D 11 matrix is the same as the matrix D 33 . The Eq. (23) is a tridiagonal block matrix.
The matrix we have generated is diagonally dominant and can solve through Thomas algorithm. Which authenticate the consistency & accuracy of the solution of the form

Description of Six order Compact Finite Difference Scheme
For complex systems the results will be dependent on the formation of the mesh, We apply higher-order compact scheme at the system in Eq. (1) with a uniform mesh at l = m = h l = h m . Scheme description is as follows:

First Boundary Point 1:
At the rst boundary point, the six order compact scheme is of the following form.
Above system in Eq. (26), the coef cients can be found by matching Taylor's series expansion comparing with various orders up to order O 7 , as a result, construction of the linear system is obtained. By constructing the linear system values of d s, which can be solved in the usual way to get the following along l direction, d 1 [36,37]. Others ones can found in the same way.   [36][37][38][39]. Others ones can found in the same way.

Nth Boundary Point:
At Nth boundary point of six order compact scheme is of the following way: Above system in Eq. (28), by constructing the linear system values of d s, which can be solved in the usual way as done in boundary point 1 and 2.

Implementation Algorithm:
By arranging Eqs. (26)- (28) in the following algorithm: where P are mentioned in Eq. (1), also matrices A and B are N m × N m sparse with triangular nature along C and D are N l × N l sparse with triangular in shape.

Theorem:
The truncation error in the compact six order nite difference scheme for equations in the system (1) is,

Error Analysis
The convergence benchmark, ef ciency and accuracy of the proposed scheme in terms of norms can be de ned as: where

Stability Analysis
The stability is concerned with the growth or decay of the error produced in the nitedifference solution. For the representation of theoretical analysis, we set P = 0 in Eq. (1). Assuming the boundary conditions are accurately propagating, we can apply the Fourier analysis method to our proposed equation.
De nition : For a time-dependent PDE, the corresponding difference scheme is stable in the norm . if there exists a constant M such that e n ≤ M e 0 , for all n∇t ≤ t F where M is independent of ∇t, ∇x and initial condition e 0 .
Following the Von Neumann stability analysis criteria, x the non-linear terms so that for linear stability, the numerical solution can be displayed in the following way: where is the amplitude at time level n, √ −1 is called the imaginary unit. l , m leads to wave number in l, and m directions with l h l , m h m are phase angles. The ampli cation factor is de ned by By using Eqs. (30) and (35) and dividing by r.h.s of Eq. (35) and simplifying, we have the following form: where I = √ −1 and Eq. (37) is of the following way: where R & S are the compact forms of Eq. (37). For stability, it has to satisfy the following condition: After simpli cation to an aforementioned condition which holds true. Therefore, |E| ≤ 1 [38][39][40][41]. Hence the scheme is unconditionally stable.

Experimental Results
The novel numerical scheme is compared with the analytical results of Eq. (1) by using tanhcoth method. For this objective, we consider the same parameter α = µ = η = 1, and varying β. Numerical and analytic solutions are compared and justi ed in term of error norms to magnify the importance of higher accuracy.
Furthermore, to avoid turbulence, by varying β values in the Tab. 1 with grid size (15 × 15), dt = 0.001 and grid space =0.3125 with respect to time = 1 is observed. Improvement in accuracy is noted by varying the values of β parameter. Also, the BH equation produced the best results by using six order compact nite difference scheme. At different β values, Tab. 2 indicates error which increased at a very low rate by changing the values of β from high to low which make the comparison to previous work give authentication for accuracy [34]. The truncation error is calculated in Tab. 3, using L 2 , Relative error and L ∞ with xed grid size (31 × 31). By changing time steps dt = 0.001 with the same grid size showed results in the Tab. 4. The approximate results using six order compact scheme correspond to error norm are shown in the Tab. 6. In this, table the comparison of fourth-order and six order are analyzed by re ning the temporal space, which shows this scheme is better than the corresponding fourth-order. In the Tab. 7 six order and fourth-order compact nite difference scheme comparison is carried out which measured in term of L ∞ norm. Different parameters are also observed under the same scheme. In the Tab. 8 scheme ef ciency encountered using L ∞ , L 2 & Relative error norms. Graphical representation of numerical schemes on BH equation is observed. Comparison of analytical and numerical results by using fourth and sixth-order compact nite difference scheme has been analyzed. At t = 2, β = 0.1, dt = 0.0001, grid = (21 × 21) can be seen from the Fig. 1. While six order scheme at β = 0.1 with time-space dt = 0.0001 and grid space (21 × 21) is seen from the Fig. 2 which shows more accurate and re ne results as compared with Fig. 1 using the same parameter. In Figs. 3 and 4, analysis shows that the error norm using fourth-order scheme at β = 0.001. While in Fig. 5, we choose β = 0.0001 using a higher-order scheme to analyze error pro le at grid size (51 × 51). In summary, it is aspirant from the gures and tables; the analytical and numerical solutions are best tted with generation encrypting. In the end, the novel six order compact scheme is the best agreement with the analytical solution.
Comparison between approximation and analytical solutions is made at the nal time of computation time = 2 s at the critical point (1, 1) using fourth-order compact scheme at grid size (15 × 15). Comparison between approximation and analytical solutions is made at the nal time of computation time = 1 at the critical point (1, 1) using fourth-order compact scheme at grid size (31 × 31).   Tab. 3 shows error pro le data by using fourth-order compact scheme at gridsize = (31 × 31) for unknown value Z(l, m, t). Selftime: is the time spent in a function excluding the time spent in its child functions while Totaltime is the time to execute the algorithm.
Tab. 4 shows error pro le data by using Six order compact scheme at gridsize = (31 × 31) for unknown value Z(l, m, t).

Central Processing Unit Performance
A combinatorial logic circuit executes the mathematical operation for each function in the algorithm within the central processing unit. To establish the platform of CPU performance along physical memory transmission capacity is observed when the higher-order compact scheme is developed by using MATLAB software [35,39,42,43]. By increasing the grid size, the number of calculations is increased, and it is dif cult to overcome such issue which can take a longer time to execute. Because of numerical schemes ef ciency, the computational experiment is done on two different computer machines like Lenovo 6th generation having 2.4 GHz 8 cores and 16 GB memory along 5th generation Dell machine having 4 physical cores and 16 logical cores. Different feathers involved in two computational experiments can be analyzed from the following data tables. Tab. 7 shows results for the different grid using 6th order compact scheme on Lenovo CPU oriented computational machine (MATLAB software).
Tab. 8 shows results for the different grid using 6th order compact scheme on DELL CPU oriented computational machine (MATLAB software).  Comparison is performed to analyze Dell with Lenovo machines with both clock rate performance and relative performance. Thus MATLAB handles problems with care, and we can analyze results at each point of the loop and any iteration during computations.

Conclusion
Higher-order schemes for determining the two dimensional Burgers Huxley equation was developed in this paper. As it was not studied before by using such schemes of diffusive dissipation of errors. We came to know that the BH equation in two dimensional which is studied to nd ef ciency, accuracy and stability and by comparing with analytical and numerical approaches in terms of L 2 , L ∞ & relative errors. It is evident from the fact that computed numerical experiments of two dimensional Burgers Huxley equation, solutions obtained by fourth and six order schemes are in good agreement with the analytical solutions. Figures and tables clearly show the tendency of fast and monotonic convergence of the results toward the analytical solution. Also, the computational discretization of the proposed model results in a sparse tridiagonal structure of the matrix, which can be overcome by the Thomas algorithm. Results lead to a remarkable improvement in accuracy, ef ciency and computer performance which can be seen from data tables.

Funding Statement:
The authors received no speci c funding for this study.

Con icts of Interest:
The authors declare that they have no con icts of interest to report regarding the present study.