The time-dependent singularly perturbed Burgers-Huxley equation models a large class of physical phenomena such as the interaction between convection effect, reaction mechanism and diffusion transport. In one dimensional space this equation has the form [1, 2]

where

α ≥ 0, β ≥ 0, γ ∈ (0, 1) are real parameters and u₀(x), h₁(t),and h₂(t) are sufficiently smooth prescribed functions. Here, 0 < ε ≪ 1 denotes the perturbation parameter.

The field of numerical methods for singularly perturbed problems has flourished significantly over the last couple of decades. The studies focused on the design and analysis of numerical methods that were parameter-uniformly convergent. In other words, research was mostly motivated by the challenge posed by the presence of the singular perturbation parameter. A vast majority of researchers discussed linear problems. Nonlinear singularly perturbed equations, and in particular Equation (1.1), received little attention from scholars in this research area. Nonlinearity constitutes an another layer of difficulties of singularly perturbed problems.

In general, the solution of the singularly perturbed Burgers-Huxley problem (1.1) with arbitrary initial and boundary conditions (IBCs) cannot be expressed in terms of a finite number of elementary functions [2, 3]. Thus, several scholars seek approximate analytic solutions using different transformation techniques [4–6]. However, these techniques still hold for only some specific parameters and initial boundary conditions. Hence, for general cases, there is a need to develop numerical methods to obtain an approximate solution of the problem.

For the last two decades, various numerical methods have been studied to solve the Burgers-Huxley equations of type (1.1) for ε = 1 and specific IBCs in the framework of NSFD methods. For example, A.R. Appadu et al. [7] developed two novel nonstandard finite difference schemes, and explicit exponential finite difference method and a fully implicit exponential finite difference method.

For the singularly perturbed case, that is when 0 < ε ≪ 1, various parameter-uniformly convergent methods have been developed to solve (1.1) [1, 2, 8, 9]. It is to be noted that all these works used the Newton's quasilinearization to deal with the nonlinearity.

In this article, a nonstandard finite difference method for the singularly perturbed case is proposed. To deal with the nonlinearity, rather than using quasilinearization, for the first time, nonlinear terms are approximated in a nonlocal manner following one of Micken's rules [10]. The resulting method preserves the properties of the continuous solution and provides reliable numerical results. The method is proved to be first order parameter-uniform convergent in time and space.

The rest of this article is structured in the following manner. In the next section, we study a priori estimates of the analytic solution to the problem. Section 3 is about the proposed NSFD method and its parameter-uniform convergence. Section 4 deals with the implementation of the method to confirm the theoretical results and compare with other methods. Section 5 concludes the present work and provides a direction for future work.

Throughout this article, we assume that u₀, h₁, and h₂ are sufficiently smooth functions and that a and b satisfy

Under these assumptions the problem (1.1) has a unique solution u(x,t)∈C2,1(D̄) which exhibits a boundary layer near x = 1 [1].

In the rest of the paper, C denotes a generic constant independent of the parameters and mesh sizes, and ||.||_D denotes the maximum norm over D. For the solution u(x, t) of (1.1), we have the following bounds.

Lemma 2.1. [Uniform stability estimate for continuous problem]

Let u(x, t) be the exact solution of (1.1) on D̄, then

where D_i = {(x, t):t = 0, x ∈ [0, 1]}, ∂D = D_i∪D_L∪D_R, D_L = {(x, t):x = 0, t ∈ [0, T]}, and D_R = {(x, t):x = 1, t ∈ [0, T]}.

Proof: The proof can be seen in [9].                      □

Remark 2.2. From our assumption in (2.1) and Lemma 2.1, one can observe that the solution u(x, t) satisfies

This section is dedicated to the construction of a new scheme and to the analysis of its parameter-uniform convergence. The first step is to discretize the solution domain D̄ and provide the definition of the NSFD methods given in [11] (a revised version of [10]).

Let M and N be positive integers and D̄M,N=Ω̄M×[0,T]N be uniform grid discretization of the solution domain D̄=Ω̄×[0,T] such that

and

where h = 1/M and Δt = T/N are spatial and temporal step sizes, respectively.

Definition 3.1. A discrete scheme to determine approximate solutions umn to the solution u(x, t) of the problem (1.1) is called a NSFD method if at least one of the following conditions is satisfied [10]:

For example, denominator functions that satisfy the above conditions are

and so on.

and so on.

Definition 3.2. Assume that the solution of (1.1) satisfies some property P. The difference equation of (1.1) in umn is called (qualitatively) stable with respect to P if, for any values of the mesh sizes Δt and h, solution of the difference equation replicates the property P.

Remark 3.3. In [10], Mickens set five rules for the constructions of the finite difference models that can replicate the properties of the exact solution. Definition 3.1 satisfies only two of these rules.

The remaining rules are expressed in terms of definition 3.2. For example, the schemes under consideration in this paper is qualitatively stable with respect to the order of the differential equation, they do satisfy positivity and uniform boundedness.

To construct the scheme, first we semidiscretize the problem (1.1) in time direction and then in space direction.

In this section, two test examples are provided to demonstrate the efficiency and applicability of the proposed numerical method. The first example is taken from a recent article [7] for ε = 1, and we modified it by multiplying the highest derivative term by ε to make the problem singularly perturbed. It is the first time to consider this example for the singularly perturbed case. The second example is taken from [2] for different initial and boundary conditions to satisfy our assumption (2.1). All the computations are carried out by Intel Coreå i5-4210M CPU @2.60GHz×4 with Matlab 2017.

Example 4.1. Consider the following singularly perturbed Burgers-Huxley problem:

where

For ε = 10⁻⁴, α = β = 1 and γ = 0.5, we plotted the numerical solution of Example 4.1 using the proposed method in Figure 1. As we see, the method resolves the boundary layer at x = 1. Thus, our proposed method replicates the property of the continuous solution.

Since the exact solution of this problem is not known, for each ε, the maximum pointwise error is calculated using double mesh principle [18] given by

where um;Mn;N and u2m;2M2n;2N are approximate solutions to problem (3.9) on D^{M, N} and D^{2M, 2N}, respectively. The corresponding order of convergence is given by

In Tables 1, 2, we compute the maximum absolute errors and the corresponding order of convergences using the proposed method for α = β = 1, γ = 0.5 and various values of ε with fixed values of M and N, respectively. From these results, we can observe that the method is ε-uniform convergent of order one both in time and space. This confirms the theoretical error results.

Table 3 compares absolute errors at some values of x and t using the proposed method and methods in [7, 19] and the nonlinear nonstandard finite difference (NNSFD) method given by

where the denominator functions

The nonlinear system (4.1) can be easily solved using Newton's method (see in [7]). The solution at the previous time-step is taken as the initial estimate. The methods considered in [7] and [19] are NSFD methods and variational iteration method (VIM), respectively. From the table, we observe that our proposed method gives more accurate results.

Also, in Table 4, we compare the CPU times required to compute the solutions of Example 4.1 for M = 10, N = 2000, α = β = 1, γ = 0.001, T = 10 using the LNSFD and NNSFD methods. As one can observe in the table, the CPU time for NNSFD is larger than the LNSFD.

Example 4.2. Consider the following SPBHE [2]:

In this paper, a nonstandard finite difference method for a singularly perturbed Burgers-Huxley equation has been developed. First, the backward-Euler scheme was applied to discretize the problem (1.1) with respect to time derivative and the upwind nonstandard finite difference scheme on uniform mesh to approximate the spatial derivative. Then, the presented method was proved to be first-order convergent in both the spatial and temporal variables. Numerical results are given in Figures 1, 2 and in Tables 1–6 for two test examples to confirm the theoretical results and to compare with recent results. It has been observed from these figures and tables that the numerical results are in agreement with the theoretical findings. For ε = 1, in Table 3, comparisons with the NNSFD and the VIM of [19] and the NSFD method of [7] reveal that the proposed NSFD method gives more accurate results. In addition, the present method is also applicable when 0 < ε < 1. Thus, in all cases, the present method produces more accurate results than the existing schemes.