Commun. Comput. Phys., 12 (2012), pp. 1417-1433. |
A Novel Numerical Method of O(h^4) for Three-Dimensional Non-Linear Triharmonic Equations R. K. Mohanty ^{1*}, M. K. Jain ^{2}, B. N. Mishra ^{3} 1 Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi-110 007, India.2 Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi-110 016, India. 3 Department of Mathematics, Utkal University, Vani Vihar, Bhubaneswar-751 004, India. Received 8 September 2010; Accepted (in revised version) 6 January 2012 Available online 8 May 2012 doi:10.4208/cicp.080910.060112a Abstract In this article, we present two new novel finite difference approximations of order two and four, respectively, for the three dimensional non-linear triharmonic partial differential equations on a compact stencil where the values of u, $\partial^2u/\partial n^2$ and $\partial^4u/\partial n^4$ are prescribed on the boundary. We introduce new ideas to handle the boundary conditions and there is no need to discretize the derivative boundary conditions. We require only 7- and 19-grid points on the compact cell for the second and fourth order approximation, respectively. The Laplacian and the biharmonic of the solution are obtained as by-product of the methods. We require only system of three equations to obtain the solution. Numerical results are provided to illustrate the usefulness of the proposed methods. AMS subject classifications: 65N06PACS: 02.60.Lj, 02.70.Bf Key words: Finite differences, three dimensional non-linear triharmonic equations, fourth order compact discretization, Laplacian, biharmonic, maximum absolute errors. *Corresponding author. Email: rmohanty@maths.du.ac.in (R. K. Mohanty), bn_misramath@hotmail.com (B. N. Mishra) |