Introduction

Consider the two-dimensional elliptic partial differential equation (1) which is defined in the solution domain Ω = {(x, y):0 < x, y < 1}, where functions D(x) and g(x, y) ∈ C²(Ω). The corresponding Dirichlet boundary conditions are given by (2) where ∂Ω is its boundary. These types of problems arise very frequently in different areas of applied mathematics and physics such as convection-diffusion equation which describes the transport phenomena, and the Poisson’s equation which is broadly used in electrostatics, mechanical engineering and theoretical physics. Thus, solving elliptic differential equation have been of interest to many authors [1–3].

In 1968, Bickley [4] suggested the use of cubic splines for solving a linear ordinary differential. Following this, Albasiny and Hoskins [5] approximated the solutions by applying the cubic spline interpolation introduced by Ahlberg et al. [6], which leads to a matrix system of tri-diagonal instead of upper Hessenberg form which was obtained by Bickley [4]. The cubic spline method suggested by Bickley [4] was then examined by Fyfe [7]. Fyfe concluded that spline method is better than the usual finite difference method in terms of its accuracy and also its flexibility to get the approximation at any point in the domain. Due to its simplicity, many researchers started to work on spline methods for solving boundary value problems [8–11]. To mention a few, Bialecki et al. [12] formulated a new fourth order one step nodal bicubic spline collocation methods for the solution of various elliptic boundary value problems. Mohanty and Gopal [13] proposed a high accuracy cubic spline finite difference approximation of O(k² + h⁴) accuracy for the solution of non-linear wave equation. Goh et al. [14] discussed the solution for one-dimensional heat and advection-diffusion by using a combination of finite difference approach and cubic B-spline method.

Over the last few decades, we have seen the formulation of group iterative methods for solving the two dimensional elliptic partial difference Eqs [15–18]. In 1991, a half-sweep iterative method had been introduced by Abdullah [19] via the explicit decoupled group (EDG) iterative method which was shown to be faster and computationally economical than the existing explicit group (EG) method due to Yousif and Evans [18] for solving elliptic partial differential equation. Inspired by Abdullah [19], Othman and Abdullah [20] proposed a quarter-sweep iteration through the modified explicit group (MEG) method. Following this, Ali and Ng [21] extended the idea and formulated the modified explicit decoupled group (MEDG) method for solving two-dimensional Poisson equation. The MEDG method exhibits a better convergence rather than the existing group schemes of the same family, namely EG, EDG and MEG methods.

In 1986, Yousif and Evans [18] developed the explicit group (EG) iterative method where a small group of 2, 4, 9, 16 and 25 points were constructed in the iterative processes for solving Laplace’s equation. The numerical results show that the EG method is simpler to program compared to the block (line) iterative methods and it requires less storage. However, this method was solely formulated using the usual standard finite difference discretization which restricts the solutions at only certain points of the solution domain. This, thus, motivate us to adopt the idea in using splines in the formulation of the group methods.

In this paper, a new method, namely spline explicit group (SEG) iterative method, which incorporates cubic spline with group iterative scheme, is developed for solving the elliptic problems. Using a cubic spline approximation in the x-direction and central difference in the y-direction, we obtain a new three level implicit nine-point compact finite difference formulation. Then, a four point explicit group iterative scheme is applied to the obtained system. The performance of the method will be investigated via two benchmark problems, that is the convection-diffusion equation and Poisson’s equation.

The Cubic Spline Approximation and Numerical Scheme

Here, the solution domain Ω = [0, 1] × [0, 1] is divided by h > 0 in x-direction and k > 0 in y-direction. Therefore, the grid points (x_l, y_m) are represented as x_l = lh and y_m = mk, l = 0, 1, …, N_x, m = 0, 1, …, N_y, where N_x and N_y are positive integers. Let U_{l, m} be the approximation solution of u_{l, m} at the grid point (x_l, y_m).

Suppose that S_m(x) is the m-th mesh row cubic spline polynomial which interpolates the value U_{l, m} at (x_l, y_m), is given by [6] (3) for x_l−1 ≤ x ≤ x_l, where l = 1, 2, …, N_x and m = 0, 1, 2, …, N_y. For each m-th mesh row, the cubic spline S_m(x) satisfies the following properties

1. S_m(x) coincides with a polynomial of degree three on each [x_l−1, x_l], l = 1, 2, …, N_x, m = 0, 1, 2, …, N_y
2. S_m(x) ∈ C²[0, 1], and
3. S_m(x_l) = U_{l, m}, l = 0, 1, 2, …, N_x, m = 0, 1, 2, …, N_y

The derivatives of cubic spline S_m(x) can be obtained as below (4) (5) And, from Eq (1), it gives (6) When x = x_l, Eq (4) becomes (7) Similarly, for x ∈ [x_l, x_l+1], it gives (8) Combining both Eqs (7) and (8), the following approximation can be obtained (9) Further, we have (10) (11) By using the continuity of first derivative at (x_l, y_m), which is, , the following relation can be obtained (12) The following approximations are considered (13a) (13b) (13c) (14a) (14b) (14c) Eqs (14b) and (14c) are obtained from the second-order one-sided finite difference scheme. For the derivatives of S_m(x), we consider (15a) (15b) (15c) (15d) (15e) (15f) By using Taylor series expansion about the grid point (x_l, y_m), Eq (1) can be written as (16) where T_{l, m} is the local truncation error. Substituting the above approximations (13)–(15) into (16), it results (17) where λ is the mesh ratio, denoted by λ = (k/h). If the singular terms like appear in the functions D(x) and/or g(x, y), which is unable to evaluate at x = 0. The following approximations are considered (18a) (18b) where Thus, neglecting the higher order terms and local truncation error, Eq (17) can be written as (19) This modified equation retains its order of accuracy everywhere throughout the solution region, moreover in the vicinity of the singularity. Note that, this proposed scheme (19) is of O(k² + k²h² + h⁴) and applicable to both singular and non-singular elliptic equations of the form (1).

Spline Explicit Group Method

We apply Eq (19) to any group of four points on the solution domain (as shown in Fig 1). Then, a (4 × 4) system as below, can be obtained (20) where and Eq (20) can be inverted and written in explicit forms (21) where and The Gauss-Seidel technique is employed to accelerate the convergence process. Iterations are generated in groups of four points over the entire spatial domain until the convergence test is satisfied. Once the approximations U_l,m had been calculated, the value of M_l,m can be easily obtained by solving the system generated by (12). Then, the piecewise polynomial of the function can be obtained from Eq (3). Finally, the approximate solution at any point at m-th mesh row can be easily calculated.

Applying Eq (21) to each of the group in natural row ordering (Fig 2) will lead to a linear system where the matrix of coefficient A is given by (22) with The submatrices are given by In order to derive the explicit formulae, the matrix A is transformed into A^E and vector b is modified into b^E, where, The block structure of A^E is the same as matrix A with the nonzero block R₀ replaced by identity matrices, I and the blocks R_i and , replaced by , i = 0, 2, 3, 4, 5, 6 and , j = 3,4,6 respectively. Since the coefficient matrix (22) is block tridiagonal with non-vanishing diagonal element, it is π-consistently ordered and has property-A^(π) [22]. Thus, the theory of block S.O.R. is also applicable to the SEG iterative method and therefore, is convergent.

Computational Complexity Analysis

In order to show the efficiency of the proposed method, computational complexity of the SEG iterative method is examined. Assume that the solution domain is discretized into even intervals, N_x and N_y in x- and y-directions, respectively. Therefore, we have (n_x − 1)(n_y − 1) grouped points and (n_x + n_y − 1) ungrouped points, where n_x = N_x − 1 and n_y = N_y − 1. This can be shown as in Fig 3.

The estimation on this computational complexity is based on the arithmetic operations performed at each iteration for the additions/substractions (Add/Sub) and multiplications/divisions (Mul/Div) operations. Therefore, the number of operations required for SEG is given as in Table 1. The total number of arithmetic operations can be obtained by multiplying the number of arithmetic operations for each iteration with the number of iterations.

Numerical Results

In this section, two benchmark test problems, whose exact solutions are known are solved by the proposed combination of cubic spline and explicit group iterative method. The results are then compared with those obtained by the

• Combination of cubic spline with block Gauss-Seidel iterative method (SBGS)
• Combination of central difference scheme with explicit group iterative method (CDEG)

where the CDEG scheme can be derived by substituting the partial derivative in Eq (1) by the central difference approximation. In all cases, we assume that u⁽⁰⁾ = 0 as the initial guess and the iterations were stopped when the estimated error was below tolerance, that is when ∣u^(s+1) − u^(s)∣ ≤ 10⁻¹² was achieved.

Conclusions

In this paper, a new method namely, the SEG iterative method was formulated for solving the elliptic boundary value problems. The presented results show that the proposed method is capable of approximating the solution very well in terms of accuracy and execution time. It can be seen that the computation cost is reduced substantially compared to those obtained by the cubic spline block Gauss-Seidel iterative method [23], especially when the grid size increases. Furthermore, in terms of accuracy, the proposed method is superior to the original central difference explicit group iterative method [18]. In conclusion, the proposed method is a viable alternative approximation tool for solving the elliptic partial differential equations.

Acknowledgments

The authors gratefully thank to the reviewers for their valuable suggestions which definitely help to improve the quality of the paper.