New Explicit Group Newton's Iterative Methods for the Solutions of Burger's Equation
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:55754", author = "Tan K. B. and Norhashidah Hj. M. Ali", title = "New Explicit Group Newton's Iterative Methods for the Solutions of Burger's Equation", 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).", volume = "5", number = "12", pages = "1934-6", }
