Scientific Research

An Academic Publisher

A New Fourth Order Difference Approximation for the Solution of Three-dimensional Non-linear Biharmonic Equations Using Coupled Approach

This paper deals with a new higher order compact difference scheme, which is, O(h

^{4}) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each internal grid point, the solution u(x,y,z) and its Laplacian Δ^{4}u are obtained. The resulting stencil algo-rithm is presented and hence this new algorithm can be easily incorporated to solve many problems. The present discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. Convergence analysis for a model problem is briefly discussed. The method is tested on three problems and compares very favourably with the corresponding second order approximation which we also discuss using coupled approach.KEYWORDS

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

R. Mohanty, M. Jain and B. Mishra, "A New Fourth Order Difference Approximation for the Solution of Three-dimensional Non-linear Biharmonic Equations Using Coupled Approach,"

*American Journal of Computational Mathematics*, Vol. 1 No. 4, 2011, pp. 318-327. doi: 10.4236/ajcm.2011.14038.

[1] | J. Smith, “The Coupled Equation Approach to the Nu- merical Solution of the Biharmonic Equation by Finite Differences,” SIAM Journal on Numerical Analysis, Vol. 7, No. 1, 1970, pp. 104-111. doi:10.1137/0707005 |

[2] | L. W. Ehrlich, “Solving the Biharmonic Equation as Cou- pled Finite Difference Equations,” SIAM Journal on Nu- merical Analysis, Vol. 8, No. 2, 1971, pp. 278-287. doi:10.1137/0708029 |

[3] | L. W. Ehrlich, “Point and Block SOR Applied to a Cou- pled Set of Difference Equations,” Computing, Vol. 12, No. 3, 1974, pp. 181-194. doi:10.1007/BF02293104 |

[4] | L. Bauer and E. L. Riess, “Block Five Diagonal Matrices and the Fast Numerical Solution of the Biharmonic Equa- tion,” Mathematics of Computation, Vol. 26, No. 118, 1972, pp. 311-326. doi:10.1090/S0025-5718-1972-0312751-9 |

[5] | R. Glowinski and O. Pironneau, “Numerical Methods for the First Biharmonic Equations and for the Two-Dimen- sional Stokes Problems,” SIAM Review, Vol. 21, No. 2, 1979, pp. 167-212. doi:10.1137/1021028 |

[6] | Y. Kwon, R. Manohar and J. W. Stephenson, “Single Cell Fourth Order Methods for the Biharmonic Equation,” Con- gress Numerantium, Vol. 34, 1982, pp. 475-482. |

[7] | J. W. Stephenson, “Single Cell Discretization of Order Two and Four for Biharmonic Problems,” Journal of Com- putational Physics, Vol. 55, No. 1, 1984, pp. 65-80. doi:10.1016/0021-9991(84)90015-9 |

[8] | R. K. Mohanty and P. K. Pandey, “Difference Methods of Order Two and Four for Systems of Mildly Non-linear Biharmonic Problems of Second Kind in Two Space Di- mensions,” Numerical Methods for Partial Differential Equations, Vol. 12, No. 6, 1996, pp. 707-717. doi:10.1002/(SICI)1098-2426(199611)12:6<707::AID-NUM4>3.0.CO;2-W |

[9] | R. K. Mohanty, M. K. Jain and P. K. Pandey, “Finite Dif- ference Methods of Order Two and Four for 2D Nonlin- ear Biharmonic Problems of First Kind,” International Journal of Computer Mathematics, Vol. 61, No. 1-2, 1996, pp. 155-163. doi:10.1080/00207169608804507 |

[10] | R. K. Mohanty and P. K. Pandey, “Families of Accurate Discretizations of Order Two and Four for 3D Mildly Nonlinear Biharmonic Problems of Second Kind,” Inter- national Journal of Computer Mathematics, Vol. 68, No. 3-4, 1998, pp. 363-380. doi:10.1080/00207169808804702 |

[11] | D. J. Evans and R. K. Mohanty, “Block Iterative Methods for the Numerical Solution of Two-dimensional Nonlin- ear Biharmonic Equations,” International Journal of Com- puter Mathematics, Vol. 69, No. 3-4, 1998, pp. 371-390. doi:10.1080/00207169808804729 |

[12] | R. K. Mohanty, D. J. Evans and P. K. Pandey, “Block Ite- rative Methods for the Numerical Solution of Three Di- mensional Mildly Nonlinear Biharmonic Problems of First Kind,” International Journal of Computer Mathe- matics, Vol. 77, No. 2, 2001, pp. 319-332. doi:10.1080/00207160108805068 |

[13] | S. Singh, D. Khattar and R. K. Mohanty, “A New Cou- pled Approach High Accuracy Numerical Method for the Solution of 2D Nonlinear Biharmonic Equations,” Neural Parallel and Scientific Computations, Vol. 17, 2009, pp. 239-256. |

[14] | D. Khattar, S. Singh and R. K. Mohanty, “A New Cou- pled Approach High Accuracy Numerical Method for the Solution of 3D Non-Linear Biharmonic Equations,” Ap- plied Mathematics and Computations, Vol. 215, No. 8, 2009, pp. 3036-3044. doi:10.1016/j.amc.2009.09.052 |

[15] | R. K. Mohanty, “A New High Accuracy Finite Difference Discretization for the Solution of 2D Non-Linear Bihar- monic Equations Using Coupled Approach,” Numerical Methods for Partial Differential Equations, Vol. 26, No. 4, 2010, pp. 931-944. doi:10.1002/num.204605 |

[16] | R. K. Mohanty, M. K. Jain and B. N. Mishra, “A Com- pact Discretization of O(h4) for Two-Dimensional Non- linear Triharmonic Equations,” Physica Scripta, Vol. 84, No. 2, 2011, pp. 025002. doi:10.1088/0031-8949/84/02/025002 |

[17] | R. K. Mohanty and S. Dey, “Single Cell Fourth Order Dif- ference Approximations for (?u/?x), (?u/?y) and (?u/?z) of the Three Dimensional Quasi-Linear Elliptic Equation,” Numerical Methods for Partial Differential Equations, Vol. 16, No. 5, 2000, pp. 417-425. doi:10.1002/1098-2426(200009)16:5<417::AID-NUM1>3.0.CO;2-S |

[18] | R. K. Mohanty, S. Karaa and U. Arora, “Fourth Order Nine Point Unequal Mesh Discretization for the Solution of 2D Non-linear Elliptic Partial Differential Equations,” Neu- ral Parallel and Scientific Computations, Vol. 14, 2006, pp. 453-470. |

[19] | R. K. Mohanty and S. Singh, “A New Highly Accurate Dis- cretization for Three Dimensional Singularly Perturbed Non-linear Elliptic Partial Differential Equations,” Nume- rical Methods for Partial Differential Equations, Vol. 22, No. 6, 2006, pp. 1379-1395. doi:10.1002/num.20160 |

[20] | L. A. Hageman and D. M. Young, “Applied Iterative Me- thods,” Dover Publications, New York, 2004. |

[21] | M. K. Jain, “Numerical Solution of Differential Equa- tions,” 2nd Edition, John Wiley, New Delhi, 1984. |

[22] | C. T. Kelly, “Iterative Methods for Linear and Non-Linear Equations,” SIAM Publications, Philadelphia, 1995. |

[23] | Y. Saad, “Iterative Methods for Sparse Linear Systems,” SIAM Publications, Philadelphia, 2003. doi:10.1137/1.9780898718003 |

[24] | G. Meurant, “Computer Solution of Large Linear Systems,” North-Holland, Amsterdam, 1999. |

[25] | W. F. Spotz and G. F. Carey, “High Order Compact Scheme for the Steady Stream-Function Vorticity Equations,” In- ternational Journal for Numerical Methods in Enginee- ring, Vol. 38, No. 20, 1995, pp. 3497-3512. doi:10.1002/nme.1620382008 |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.