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

Ranjan Kumar Mohanty, Mahinder Kumar Jain, Biranchi Narayan Mishra

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**DOI: **10.4236/ajcm.2011.14038
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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

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

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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.

Conflicts of Interest

The authors declare no conflicts of interest.

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