On the Behavior of Combination High-Order Compact Approximations with Preconditioned Methods in the Diffusion-Convection Equation

ABSTRACT

In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite difference equations are solved by preconditioned Krylov subspace methods. Numerical results are given to verify the behavior of high-order compact approximations in combination preconditioned methods for stability, convergence. Also, the accuracy and efficiency of the proposed scheme are considered.

In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite difference equations are solved by preconditioned Krylov subspace methods. Numerical results are given to verify the behavior of high-order compact approximations in combination preconditioned methods for stability, convergence. Also, the accuracy and efficiency of the proposed scheme are considered.

KEYWORDS

Compact High-Order Approximation, Diffusion-Convection Equation, Krylov Subspace Methods, Preconditioner

Compact High-Order Approximation, Diffusion-Convection Equation, Krylov Subspace Methods, Preconditioner

Cite this paper

nullA. Golbabai and M. Molavi-Arabshahi, "On the Behavior of Combination High-Order Compact Approximations with Preconditioned Methods in the Diffusion-Convection Equation,"*Applied Mathematics*, Vol. 2 No. 12, 2011, pp. 1462-1468. doi: 10.4236/am.2011.212208.

nullA. Golbabai and M. Molavi-Arabshahi, "On the Behavior of Combination High-Order Compact Approximations with Preconditioned Methods in the Diffusion-Convection Equation,"

References

[1] L. C. Evans, “Partial Differential Equations,” American Mathematical Society Providence, Rhode Island, 1999.

[2] A. Golbabai and M. M. Arabshahi, “A Numerical Method for Diffusion-Convection Equation Using High-Order Difference Schemes,” Computer Physics Communications, Vol. 181, No. 7, 2010, pp. 1224-1230. doi:10.1016/j.cpc.2010.03.008

[3] A. Golbabai and M. M. Arabshahi, “On the Behavior of High-Order Compact Approximations in One Dimensional Sine-Gordon Equation”, Physica Scripta, Vol. 83, No. 1, 2011, Article ID 015015. doi:10.1088/0031-8949/83/01/015015

[4] L. C. Evans, “Partial Differential Equations,” American Mathematical Society Providence, Rhode Island, 1999.

[5] S. Sundar and B. K. Bhagavan, “CGS, Comparison of Krylov Subspace Methods with Preconditioning Techniques for Solving Boundary Value Problems,” Computers and Mathematics with Applications, Vol. 38, No. 11-12, 1999, pp. 197-206. doi:10.1016/S0898-1221(99)00298-9

[6] A. M. Bruaset, “A Survey of Preconditioned Iterative Methods,” Longman Scientific and Technical, UK, 1995.

[7] K. J. Hout and B. D. Welfert, “Unconditional Stability of Second-Order ADI Schemes Applied to Multi-Dimensional Diffusion Equations with Mixed Derivative Terms,” Applied Numerical Mathematics, 2008, Article in Press.

[8] S. Ma and Y. Saad, “Block-ADI Preconditioners for Solving Sparse Non-Symmetric Linear Systems of Equations,” Numerical Linear Algebra, 1993, pp. 165-178.

[9] M. Bhuruth and D. J. Evans, “Block Alternating Group Explicit Preconditioning (BLAGE) for a Class of Fourth-Order Difference Schemes,” International Journal of Computer Mathematics, Vol. 63, No. 1-2, 1997, pp. 121-136. doi:10.1080/00207169708804555

[10] M. K. Jain, R. K. Jain and R. K. Mohanty, “Fourth-Order Finite Difference Method for 2-D Parabolic Partial Differential Equations with Non-Linear First-Derivative Terms,” Numerical Methods for Partial Differential Equations, Vol. 8, No. 1, 1992, pp. 21-31. doi:10.1002/num.1690080102

[11] G. I. Shishkin and L. P. Shishkina, “A Higher Order Richardson Scheme for a Singularly Perturbed Semilinear Elliptic Convection-Diffusion Equation,” Computational Mathematics and Mathematical Physics, Vol. 50, No. 3, 2010, pp. 437-456. doi:10.1134/S0965542510030061

[12] Y. Zhang, “Matrix Theory Basic Results and Techniques,” Springer, Berlin, 1999.

[13] R. Barrett, et al., “Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods,” Society for Industrial and Applied Mathematics (SIAM), 1994, pp. xvii+118. doi:10.1137/1.9781611971538

[14] Y. Saad, “Iterative Methods for Sparse Linear Systems,” Second Edition, PWS Publishing Company, Boston, 2000.

[15] H. A. Van der Vorst, “Iterative Krylov Subspace Methods for Large Linear Systems,” Cambridge University Press, Cambridge, 2003. doi:10.1017/CBO9780511615115

[16] O. Axelsson, “Iterative Solution Methods,” Cambridge University Press,” New York, 1996.

[17] M. H. Koulaei and F. Toutounian, “On Computing of Block ILU Preconditioner for Block Tri-Diagonal Systems,” Journal of Computational and Applied Mathematics, Vol. 202, No. 2, 2007, pp. 248-257. doi:10.1016/j.cam.2006.02.029

[18] R.C. Mittal and A.H. Al-Kurdi, “An Efficient Method for Constructing an ILU Preconditioner for Solving Large Sparse Non-Symmetric Linear Systems by the GMRES Method,” Computers and Mathematics with Applications, Vol. 45, 2003, pp. 1757-1772. doi:10.1016/S0898-1221(03)00154-8

[19] D. W. Peaceman and H. H. Rachford, “The Numerical Solution of Parabolic and Elliptic Differential Equations,” Journal of the Society for Industrial and Applied Mathematics, Vol. 3, No. 1, 1955, pp. 28-41. doi:10.1137/0103003

[20] D. M. Young, “Iterative Solution of Large Linear Systems,” Academic Press, New York, 1971.

[21] R. S. Varga, “Matrix Iterative Analysis,” Prentice Hall, Englewood Cliffs, 1962.

[22] D. J. Evans and W. S. Yousif, “The Block Alternating Group Explicit Method (BLAGE) for the Solution of Elliptic Difference Equations,” International Journal of Computer Mathematics, Vol. 22, No. 2, 1987, pp. 177- 185. doi:10.1080/00207168708803590

[23] R. K. Mohanty, “Three-Step BLAGE Iterative Method for Two-Dimensional Elliptic Boundary Value Problems with Singularity,” International Journal of Computer Mathematics, Vol. 84, No. 11, 2007, pp. 1613-1624. doi:10.1080/00207160600825205

[24] D. J. Evans and M. Sahimi, “The Alternating Group Explicit (AGE) Iterative Method to Solve Parabolic and Hyperbolic Partial Differential Equations,” Annual Review of Numerical Fluid Mechanics and Heat Transfer, Vol. 11, 1989, pp. 283-390.

[1] L. C. Evans, “Partial Differential Equations,” American Mathematical Society Providence, Rhode Island, 1999.

[2] A. Golbabai and M. M. Arabshahi, “A Numerical Method for Diffusion-Convection Equation Using High-Order Difference Schemes,” Computer Physics Communications, Vol. 181, No. 7, 2010, pp. 1224-1230. doi:10.1016/j.cpc.2010.03.008

[3] A. Golbabai and M. M. Arabshahi, “On the Behavior of High-Order Compact Approximations in One Dimensional Sine-Gordon Equation”, Physica Scripta, Vol. 83, No. 1, 2011, Article ID 015015. doi:10.1088/0031-8949/83/01/015015

[4] L. C. Evans, “Partial Differential Equations,” American Mathematical Society Providence, Rhode Island, 1999.

[5] S. Sundar and B. K. Bhagavan, “CGS, Comparison of Krylov Subspace Methods with Preconditioning Techniques for Solving Boundary Value Problems,” Computers and Mathematics with Applications, Vol. 38, No. 11-12, 1999, pp. 197-206. doi:10.1016/S0898-1221(99)00298-9

[6] A. M. Bruaset, “A Survey of Preconditioned Iterative Methods,” Longman Scientific and Technical, UK, 1995.

[7] K. J. Hout and B. D. Welfert, “Unconditional Stability of Second-Order ADI Schemes Applied to Multi-Dimensional Diffusion Equations with Mixed Derivative Terms,” Applied Numerical Mathematics, 2008, Article in Press.

[8] S. Ma and Y. Saad, “Block-ADI Preconditioners for Solving Sparse Non-Symmetric Linear Systems of Equations,” Numerical Linear Algebra, 1993, pp. 165-178.

[9] M. Bhuruth and D. J. Evans, “Block Alternating Group Explicit Preconditioning (BLAGE) for a Class of Fourth-Order Difference Schemes,” International Journal of Computer Mathematics, Vol. 63, No. 1-2, 1997, pp. 121-136. doi:10.1080/00207169708804555

[10] M. K. Jain, R. K. Jain and R. K. Mohanty, “Fourth-Order Finite Difference Method for 2-D Parabolic Partial Differential Equations with Non-Linear First-Derivative Terms,” Numerical Methods for Partial Differential Equations, Vol. 8, No. 1, 1992, pp. 21-31. doi:10.1002/num.1690080102

[11] G. I. Shishkin and L. P. Shishkina, “A Higher Order Richardson Scheme for a Singularly Perturbed Semilinear Elliptic Convection-Diffusion Equation,” Computational Mathematics and Mathematical Physics, Vol. 50, No. 3, 2010, pp. 437-456. doi:10.1134/S0965542510030061

[12] Y. Zhang, “Matrix Theory Basic Results and Techniques,” Springer, Berlin, 1999.

[13] R. Barrett, et al., “Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods,” Society for Industrial and Applied Mathematics (SIAM), 1994, pp. xvii+118. doi:10.1137/1.9781611971538

[14] Y. Saad, “Iterative Methods for Sparse Linear Systems,” Second Edition, PWS Publishing Company, Boston, 2000.

[15] H. A. Van der Vorst, “Iterative Krylov Subspace Methods for Large Linear Systems,” Cambridge University Press, Cambridge, 2003. doi:10.1017/CBO9780511615115

[16] O. Axelsson, “Iterative Solution Methods,” Cambridge University Press,” New York, 1996.

[17] M. H. Koulaei and F. Toutounian, “On Computing of Block ILU Preconditioner for Block Tri-Diagonal Systems,” Journal of Computational and Applied Mathematics, Vol. 202, No. 2, 2007, pp. 248-257. doi:10.1016/j.cam.2006.02.029

[18] R.C. Mittal and A.H. Al-Kurdi, “An Efficient Method for Constructing an ILU Preconditioner for Solving Large Sparse Non-Symmetric Linear Systems by the GMRES Method,” Computers and Mathematics with Applications, Vol. 45, 2003, pp. 1757-1772. doi:10.1016/S0898-1221(03)00154-8

[19] D. W. Peaceman and H. H. Rachford, “The Numerical Solution of Parabolic and Elliptic Differential Equations,” Journal of the Society for Industrial and Applied Mathematics, Vol. 3, No. 1, 1955, pp. 28-41. doi:10.1137/0103003

[20] D. M. Young, “Iterative Solution of Large Linear Systems,” Academic Press, New York, 1971.

[21] R. S. Varga, “Matrix Iterative Analysis,” Prentice Hall, Englewood Cliffs, 1962.

[22] D. J. Evans and W. S. Yousif, “The Block Alternating Group Explicit Method (BLAGE) for the Solution of Elliptic Difference Equations,” International Journal of Computer Mathematics, Vol. 22, No. 2, 1987, pp. 177- 185. doi:10.1080/00207168708803590

[23] R. K. Mohanty, “Three-Step BLAGE Iterative Method for Two-Dimensional Elliptic Boundary Value Problems with Singularity,” International Journal of Computer Mathematics, Vol. 84, No. 11, 2007, pp. 1613-1624. doi:10.1080/00207160600825205

[24] D. J. Evans and M. Sahimi, “The Alternating Group Explicit (AGE) Iterative Method to Solve Parabolic and Hyperbolic Partial Differential Equations,” Annual Review of Numerical Fluid Mechanics and Heat Transfer, Vol. 11, 1989, pp. 283-390.