SOLUTION OF CONVECTION-DIFFUSION PROBLEMS USING FOURTH ORDER ADAPTIVE CUBIC SPLINE METHOD

In this paper, using adaptive cubic spline, we have suggested a numerical scheme for solving a convectiondiffusion problem having layer structure. The numerical scheme is derived with this spline and non-standard finite differences of the first derivative. The tridiagonal solver is used to solve the system of the numerical method. The analysis of convergence of the method is briefly discussed and the fourth order is shown. The numerical results of the examples were tabulated and compared to the existing computational results in order to support the higher accuracy of the proposed numerical scheme.


INTRODUCTION
It is well known that many physical problems with many small parameters often involve the 818 K. MAMATHA AND K. PHANEENDRA solution of boundary value problems. This paper deals with convection-diffusion boundary value problems involving small parameter. These problems are characterized by the inclusion of a small perturbation parameters which multiply the second order derivative. In many fields of engineering and science, such types of problems exist such as chemical reactor theory, transport phenomena in chemistry, lubrication theory and biology.
A broad verity of books has been found in the literature for the convection -diffusion problems or singular perturbation problems (SPPs) [2,3,4,10,14]. One can refer a book on splines by Micula [9]. The survey papers [6,8] provides a detailed research work on SPP problems. In [1], the authors suggested a difference schemes of second and fourth order based on cubic spline in compression for SPP. A variable-mesh second-order difference scheme via cubic splines is proposed to solve SPP in [5]. In the paper [7], authors used the artificial viscosity in B-spline collocation method to capture the layer behaviour of the problem. Phaneendra and Lalu [12] derived numerical scheme using Gaussian quadrature for the solution of SPP with one end layer, dual layer and internal layer. The authors in [13] extended the Numerov scheme to the SPP with first order derivative. Soujanya et al.
[15] introduced a scheme having a fitting factor in Dahlquist scheme to get the solution of SPP having dual layers. Uniform difference schemes based on a class of splines are proposed by Stojanovic [16] for the solution of non-self-adjoint SPP.
In this paper, we present a fourth order finite difference method using adaptive cubic spline to solve singularly perturbed boundary value problems. We introduce a new parameter in the difference scheme to achieve fourth order convergence for the proposed problem. The paper is organized as follows: In section 2, Description of the problem along with conditions for layer behavior is given. In section 3, we define the adaptive spline function. In section 4, we describe the numerical method for solving singularly perturbed boundary value problems, in Section 5, the truncation error and classification of various orders of the proposed method are given. In section 6, we discuss the convergence analysis of the method. Finally, numerical results and comparison with other methods are given in section 7.

DESCRIPTION OF THE METHOD
To describe the method, we considered convection-diffusion boundary value problem of the type: with boundary conditions with perturbation parameter 0 < << 1. If is a negative constant such that such that ( ) ≤ < 0 over the domain [ , ], then the layer is in the range of = .

ADAPTIVE SPLINE
With grid points in [ , ], consider the mesh such that : = 0 < 1 <⋅⋅⋅< = , where where The spline function ( , ) on [ , +1 ] is acquired with replacing i by ( + 1) in Eq. (6) and utilizing the first or second derivative continuity condition of ( , ) at = , we get the following relationship: Further relations are given below for the adaptive splines We also obtain, Remark: In the limiting case when → 0, we have

DESCRIPTION OF THE NUMERICAL PROCEDURE
At the mesh point , the suggested approach can be discretized by the convection-diffusion equation Eq. (1) as The above equations shall be replaced by Eq. (8) and using the following approximations for the first order derivative of x at the mesh points 1 , 2 , … , −1 .

TRUNCATION ERROR
Developed local truncation error associated with the scheme in Eq. (10) is Thus, for different values of ̃2 , ̃3 ,̃1 +̃4 in the scheme Eq. (10), indicates different orders: , fourth order method is derived.

CONVERGENCE ANALYSIS
The convergence analysis of the method described in previous section for the problem Eq. (1) is now being considered. The system of equations Eq. (10) in the matrix form with the boundary conditions is obtained as   Table 1 and layer structure is pictured in Figure 1. . The results are shown in Table 2 and layer behaviour is pictured in Figure 2. ) .

Its exact solution is given by
The numerical results are tabulated in Table 3 and layer profile is displayed in Figure 3. The maximum errors in solution of Example 4 are pictured in Table 4 and layer profile is displayed in Figure 4. The exact solution of this problem is .
This problem has an internal layer at = 1 2 . The numerical results are shown in the Table 6 and layer structure is pictured in Figure 6. This problem exhibits dual layers at = 0and = 1.
The exact solution is given by ( ) = ( − 2 (1 − )/ ). Table 7 shows the maximum errors in the solution. The maximum absolute errors are posed in Table 8 for different values of and h.

CONCLUSION
In this paper, we demonstrated a numerical scheme to solve a convection-diffusion problem using adaptive cubic spline. We introduce a new parameter in the difference scheme to achieve fourth order convergence for the suggested problem. We have obtained a three-term relation with the help of difference scheme which involves a parameter . The tridiagonal scheme obtained by the method is solved using the discrete invariant imbedding algorithm.
The convergence of the method has been discussed.  Tables 1-7 to illustrate the efficiency of the method and to support the method. The layer profile is pictured in Figures 1-7. We noticed that the proposed fourth order scheme has been found to produce better results.