Fitted numerical method for singularly perturbed Burger–Huxley equation

This paper deals with the numerical treatment of a singularly perturbed unsteady Burger–Huxley equation. We linearize the problem using the Newton–Raphson–Kantorovich approximation method. We discretize the resulting linear problem using the implicit Euler method and specially ﬁtted ﬁnite diﬀerence method for time and space variables, respectively. We provide the stability and convergence analysis of the method, which is ﬁrst-order parameter uniform convergent. We present several model examples to illustrate the eﬃciency of the proposed method. The numerical results depict that the present method is more convergent than some methods available in the literature. MSC: 35B25; 35K67; 65M12

In [6][7][8][9][10][11][12][13][14][15][16] and references therein, the authors constructed various analytical and numerical methods for the Burger equations. The Burger-Huxley equation, in which the highest order derivative term is affected by a small parameter ε (0 < ε 1), is classified as the singularly perturbed Burger-Huxley equation (SPBHE). The presence of ε and the nonlinearity in the problem lead to severe difficulties in approximating the solution of the problem. For instance, due to the presence of ε, the solution reveals boundary/sharp interior layer(s), and it is tough to find a stable numerical approximation. While solving SPBHE, unless specially designed meshes are used, the presented methods in [6][7][8][9][10][11][12][13][14][15][16] and other standard numerical methods fail to give acceptable results. This limitation of the conventional numerical methods has encouraged researchers to develop robust numerical techniques that perform well enough independently of ε. Kaushik and Sharma [17] investigated problem (1) using the finite difference method (FDM) on the piecewise uniform Shishkin mesh. In [18] a monotone hybrid finite difference operator on a piecewise uniform Shishkin mesh is employed to find the approximate solution for problem (1). An upwind FDM on an adaptive nonuniform grid to find an approximate solution for problem (1) is suggested by Liu et al. [19]. In [20][21][22][23] the authors proposed a parameter uniform numerical method based on fitted operator techniques for problem (1).
However, the development of the solution methodologies for problem (1) is at an infant stage. This limitation motivated us to construct a parameter uniform numerical scheme for solving problem (1) based on the fitted operator approach. The proposed method is an ε-uniformly convergent numerical algorithm that does not require a priori knowledge of the location and breadth of the boundary layer(s), which in turn increases the difficulty of finding the free oscillation solution of the problem under consideration. Also, the proposed method requires less computational effort to solve the families of the problem under consideration.
To linearize the semilinear term of Eq. (1), we choose a reasonable initial approximation y 0 (s, t) for the function y(s, t) in the term g(s, t, y(s, t), ∂y ∂s (s, t)) that satisfies both initial and boundary conditions and is obtained by the separation-of-variables method of the homogeneous part of the problem under consideration; it is given by y 0 (s, t) = y 0 (s) exp -π 2 t . Now we apply the Newton-Raphson-Kantorovich approximation technique to the nonlinear term g(s, t, y(s, t), ∂y ∂s (s, t)) of Eq. (2), which can be linearized as where {y (m) } ∞ m=0 is a sequence of approximate solutions of g(s, t, y (m) (s, t), ∂y (m) ∂s (s, t)). For simplicity, we denote y (m+1) =ŷ and substitute Eq. (3) into Eq. (2), which yields , v(s, t) = g s, t, y (m) (s, t), ∂s (s,t)) .
and the global error estimate in the temporal direction is given by

where C is a positive constant independent of ε and τ .
Proof See [23].

Spatial semidiscretization
In this section, we use the finite difference method for the spatial discretization of problem (6) with a uniform step size. For right boundary layer problem, by the theory of singular perturbations [24] the asymptotic solution of the zeroth-order approximation for problem (6) is given as where Y j+1 0 (s) is the solution of the reduced problem Taking the first terms in Taylor's series expansion for γ (s) about the point 1, Eq. (7) becomes Now we divide the interval [0, 1] into N equal parts with = 1/N yielding a space mesh where ρ = ε 2 .
Using the above expressions in Eq. (20), we get On simplifying, we get which is the required value of the constant fitting factor σ (ρ). Finally, using Eq. (19) and the value of σ (ρ) given by Eq. (21), we get where For sufficiently small mesh sizes, the above matrix is nonsingular, and |χ c i | ≥ |χ c i | + |χ + i |. Hence by [26] the matrix χ is an M-matrix and has an inverse. Therefore Eq. (22) can be solved by the matrix inverse with given boundary conditions.
Hence Lemma 4.2 confirms that the discrete scheme (22) is uniformly stable in supremum norm.

Lemma 4.3 If Y ∈ C 3 (I), then the local truncation error in space discretization is given as
Proof By definition Using relation (22) with W = γ (0) 2 coth( γ (1)ρ 2 ), we get Thus we obtain the desired result. be the solution of the discrete problem (22). Then we have the following estimate: Proof Rewrite Eq. (22) in matrix vector form as where Z = (χ i,j ), 0 ≤ j ≤ M -1, 1 ≤ i ≤ N -1, is the tridiagonal matrix with and H = (μ We also have where Y = (Y 0 , Y 1 , . . . , Y N ) t and ( ) = ( 1 ( ), 2 ( ), . . . , N ( )) t are the actual solution and the local truncation error, respectively. From Eqs. (23) and (24) we get Then Eq. (25) can be written as where E = Y -Y = ( 0 , 1 , 2 , . . . , N ) t . Let S be the sum of elements of the ith row of Z. Then we have ). Since 0 < ε 1, for sufficiently small , the matrix Y is irreducible and monotone. Then it follows that Z -1 exists and its elements are nonnegative [27]. Hence from Eq. (26) we obtain Let χ ki be the (ki)th element of Z -1 . Since χ ki ≥ 0, by the definition of multiplication of matrices with its inverses we have Therefore it follows that for some i0 between 1 and N -1, and B i0 = i . From equations (23), (27), and (28) we obtain which implies This implies that the spatial semidiscretization process is convergent of second order. (1), and let Y j i be the numerical solution obtained by the proposed scheme (22). Then we have the following error estimate for the totally discrete scheme:

Theorem 4.5 Let y(s, t) be the solution of problem
Proof By combining the result of Lemmas 3.1 and 4.4 we obtain the required bound.

Numerical examples, results, and discussion
In this section, we consider three model problems to verify the theoretical findings of the proposed method. As the exact solutions of the considered examples are not known, we calculate the maximum absolute error for each ε given in [28] by and the corresponding order of convergence for each ε by For all N and τ , the ε-uniform maximum error and the corresponding ε-uniform order of convergence are calculated using and r N,τ = log 2 E N,τ /E 2N,τ /2 , respectively.

Conclusion
We have presented a parameter uniform numerical scheme for the singularly perturbed unsteady Burger-Huxley equation. The developed scheme constitutes the implicit Euler in the time direction and specially fitted finite difference method in the space direction. Theoretical and numerical stability and parameter uniform convergence analysis of the developed scheme is presented. The presented method is shown to be ε-uniformly convergent with convergence order O(τ + 2 ). Several model examples are presented to illustrate the efficiency of the proposed method. The proposed scheme gives more accurate numerical results than those in [17][18][19]23].