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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.

has been cited by the following article:

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

  AUTHORS: Ranjan Kumar Mohanty, Mahinder Kumar Jain, Biranchi Narayan Mishra

  KEYWORDS: Three-Dimensional Non-Linear Biharmonic Equation, Finite Differences, Fourth Order Accuracy, Compact Discretization, Block-Block-Tridiagonal, Tangential Derivatives, Laplacian, Stream Function; Reynolds Number

  JOURNAL NAME: American Journal of Computational Mathematics, Vol.1 No.4, December 9, 2011

  ABSTRACT: This paper deals with a new higher order compact difference scheme, which is, O(h4) 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 Δ4u 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.