**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**31598

##### New Explicit Group Newton's Iterative Methods for the Solutions of Burger's Equation

**Authors:**
Tan K. B.,
Norhashidah Hj. M. Ali

**Abstract:**

In this article, we aim to discuss the formulation of two explicit group iterative finite difference methods for time-dependent two dimensional Burger-s problem on a variable mesh. For the non-linear problems, the discretization leads to a non-linear system whose Jacobian is a tridiagonal matrix. We discuss the Newton-s explicit group iterative methods for a general Burger-s equation. The proposed explicit group methods are derived from the standard point and rotated point Crank-Nicolson finite difference schemes. Their computational complexity analysis is discussed. Numerical results are given to justify the feasibility of these two proposed iterative methods.

**Keywords:**
Standard point Crank-Nicolson (CN),
Rotated point Crank-Nicolson (RCN),
Explicit Group (EG),
Explicit Decoupled Group (EDG).

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1063352

**References:**

[1] K. B. Tan, N. H. M. Ali and C-H. Lai, "Explicit Group Methods in the Solution of the 2-D Convertion-Diffusion Equations," Proceedings of the World Congress on Engineering 2010, pp. 1799 - 1804.

[2] W. S. Yousif, and D. J. Evans, "Explcit Group Over-Relaxation Methods for Solving Elliptic Partial Differential Equations," Mathematics and Computers in Simulation, vol. 28, pp. 453 - 466, 1986.

[3] A. R. Abdullah, "The Four Point Explicit Decoupled Group EDG Method: A Fast Poisson Solver," International Journal of Computer Mathematics, vol. 38, pp. 61 - 70, 1991.

[4] W. S. Yousif, and D. J. Evans, "Explicit De-coupled Group Iterative Methods and Their Parallel Implementations," Parallel Algorithms and Applications, vol. 7, pp. 53 - 71, 1995.

[5] M. Othman, and A. R. Abdullah, "An Efficient Four Points Modified Explicit Group Poisson Solver," International Journal of Computer Mathematics, vol. 76(2), pp. 203 - 217, 2000.

[6] N. H. M. Ali, The Design and Analysis of Some Parallel Algorithms for the Iterative Solution of Partial Differential Equations. PhD Thesis, Fakulti Teknologi dan Sains Maklumat, Universiti Kebangsaan Malaysia, 1998, pp. 160 - 216.

[7] N. H. M. Ali, and K. F. Ng, "Modified Explicit Decoupled Group Method in The Solution of 2-D Elliptic PDES," Proceedings of the 12th WSEAS International Conference on Applied Mathematics, pp. 162 - 167, 2007.

[8] J. M. Burger, "A mathematical model illustrating the theory of turbulence," Advances in Applied Mechanics, vol. 1, pp. 171 - 199, 1948.

[9] J. D. Cole, "On a quasilinear parabolic equations occurring in aerodynamics," Quarterly of Applied Mathematics, vol. 9, pp. 225 - 236, 1951.

[10] D. J. Evans, "Iterative methods for solving non-linear two point boundary value problems," International Journal of Computer Mathematics, Vol. 72, pp. 395 - 401, 1999.

[11] W. Liao, "A fourth-order finite-difference method for solving the system of two-dimensional Burgers- equations," International Journal for Numerical Methods in Fluids, vol. 64, pp. 565 - 590, 2010.