A Single Sweep AGE Algorithm on a Variable Mesh Based on Off-Step Discretization for the Solution of Nonlinear Burgers ’ Equation

We discuss a new single sweep alternating group explicit iterationmethod, alongwith a third-order numericalmethod based on offstep discretization on a variable mesh to solve the nonlinear ordinary differential equation y󸀠󸀠 = f(x, y, y󸀠) subject to given natural boundary conditions. Using the proposed method, we have solved Burgers’ equation both in singular and nonsingular cases, which is the main attraction of our work. The convergence of the proposed method is discussed in detail. We compared the results of the proposed iteration method with the results of the corresponding double sweep alternating group explicit iteration methods to demonstrate computationally the efficiency of the proposed method.

These conditions ensure that the boundary value problem (1) and (2) possesses a unique solution (see Keller [1]).
With the advent of parallel computers, scientists are focusing on developing finite difference methods with the property of parallelism.Working on this, in the early 1980s, Evans [2,3] introduced the Group Explicit methods for large linear system of equations.Further he discussed the Alternating Group Explicit (AGE) method to solve periodic parabolic equations in a coupled manner.Mohanty and Evans applied AGE method along with various high order methods [4,5] for the solution of two-point boundary value problems.Later, Sukon and Evans [6] introduced a Two-parameter Alternating Group Explicit (TAGE) method for the two-point boundary value problem with a lower order accuracy scheme.In 2003 Mohanty et al. [7] discussed the application of TAGE method for nonlinear singular two point boundary value problems using a fourth-order difference scheme.In 1990, Evans introduced the Coupled Alternating Group Explicit method [8] and applied it to periodic parabolic equations.Many scientists are applying these parallel algorithms to solve ordinary and partial differential equations [9][10][11].
Recently, Mohanty [12] has proposed a high order variable mesh method for nonlinear two-point boundary value problem.Mohanty and Khosla [13,14] also devised a new thirdorder accurate arithmetic average variable mesh method for
The new third-order method is described as follows.

Application to Singular Problems
3.1.Linear Singular Problems.We now discuss the application of the proposed numerical method (5) to the linear differential equation with variable coefficients where () represents a forcing function.
For () = −/ and for  = 1 and 2, the equation above represents singular equation in cylindrical and spherical symmetry, respectively.
Let us denote Therefore applying the method (5) to the differential equation ( 6) and neglecting error term, we obtain a linear difference equation of the form where The linear difference scheme (8) has a local truncation error of (ℎ 5  ) and is free from the terms 1/ ±1 and therefore can very easily be solved for  = 1(1) in the region  ∈ (0, 1).

Nonlinear Singular
Problems.Now we consider the application to nonlinear differential equation (1).Neglecting the error term, we may rewrite the nonlinear difference equation (5) as Let We denote The Jacobian (y) may be represented as

Single Sweep AGE Method
4.1.Description of the Method.In this section we discuss the single sweep AGE iteration method.Using the boundary conditions  0 = ,  +1 = , the linear difference equation ( 8) may be written in the matrix form as where ( To implement the single sweep AGE iterative method, we split the coefficient matrix A into two submatrices A = G 1 + G 2 , where G 1 and G 2 satisfy the following conditions. (i) G 1 +  1 I and G 2 +  2 I are nonsingular for suitable choice of  1 > 0 and  2 > 0.
We will be concerned here with the situation where G 1 and G 2 are small (2 × 2) block systems.Now we discuss the case when  is even (with  0 = 0,  +1 = 1). Let So that the system ( 14) can be rewritten as Then a two-parameter AGE method for solving the afore mentioned system may be written as where z () is an intermediate vector.
Eliminating z () and combining ( 18) and ( 19), we obtain the iterative method or where The new iterative method (20) or ( 21) is called the twoparameter single sweep AGE iterative method and the matrix T  is called the iteration matrix.Now we discuss the single sweep AGE algorithm, when  is even.
For simplicity, we denote and we define By carrying out the necessary algebra in (20), we obtain the following algorithm.
For  = 1, For and then Finally, for  = , Similarly, we can write the single sweep AGE algorithm when  is odd.

Convergence Analysis. The single sweep AGE iteration method is given by
or where The matrix T  is called the iteration matrix.
To prove the convergence of the method, we need to prove that (T  ) ≤ 1, where (T  ) denotes the spectral radius of T  .Lemma 1.Let ℎ  be sufficiently small.Then, the eigenvalues of G 1 and G 2 are all real. Proof.Consider = −1 +  (ℎ  ) < 0, for sufficiently small ℎ  .
Let   ,  = 1(1), be the eigenvalues of G 1 .Then   's are the roots of the quadratic equation Simplifying, we get The discriminants of the quadratic equations are Hence, the eigenvalues of G 1 are real.
In a similar manner, we can show that the eigenvalues of G 2 are real.Now we give the sufficient condition for the convergence of the method.Theorem 2. Let   and   ,  = 1(1), be the eigenvalues of G 1 and G 2 , respectively.If then the iterative method is convergent for the system (14).
Hence, the convergence of the method (20) follows.
Similarly, we can write the single sweep Newton-AGE algorithm when  is odd.

Results and Observations
We have applied the methods to the following three examples, whose exact solutions are known to us, and have compared the results with the corresponding double sweep AGE and Newton-AGE method [14].For single sweep Newton-AGE method, we use the technique given by Evans [16].The right hand side function and boundary conditions may be obtained using the exact solutions.Here, we have taken   =  =  constant,  = 1(1) + 1.The value of the first mesh spacing on the left is given by Therefore, given the value of  and , we can calculate ℎ 1 from the above relation and the remaining mesh points are determined by ℎ +1 = ℎ  ,  = 1(1).The initial vector y (0) = 0 is used in all iterative methods and the iterations were stopped when the absolute error tolerance | (+1) −  () | ≤ 10 −10 was achieved.

Discussion and Conclusion
We have discussed a new single sweep AGE iterative method and three-point off-step method of accuracy (ℎ  three examples including two nonlinear and singular cases are presented.The results obtained are compared with the corresponding double sweep AGE method and show superiority over the latter.The double sweep AGE method requires two sweeps to solve a problem, whereas the single sweep AGE method requires only one sweep to solve the problem.Experimentally, as compared to the double sweep method the corresponding single sweep method requires much less number of iterations as it uses less intermediate variables.The method can be extended to solve multidimensional problems and is suitable for use on parallel computers.
3 ) on a variable mesh for the solution of nonlinear two point boundary value problems.To demonstrate the efficiency of the method,