A new coupled approach high accuracy numerical method for the solution of 3D non-linear biharmonic equations Dinesh Khattar a , Swarn Singh b , R.K. Mohanty c, * a Department of Mathematics, Kirori Mal college, University of Delhi, Delhi, India b Department of Mathematics, Sri Venkateswara college, University of Delhi, Delhi, India c Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, India article info Keywords: Finite differences Arithmetic average discretization Three-dimensional non-linear biharmonic equations Laplacian High accuracy Compact approximation Maximum absolute errors abstract In this paper, we derive a new fourth order finite difference approximation based on arith- metic average discretization for the solution of three-dimensional non-linear biharmonic partial differential equations on a 19-point compact stencil using coupled approach. The numerical solutions of unknown variable uðx; y; zÞ and its Laplacian r 2 u are obtained at each internal grid point. The resulting stencil algorithm is presented which can be used to solve many physical problems. The proposed method allows us to use the Dirichlet boundary conditions directly 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. The new method is tested on three problems and the results are compared with the corresponding second order approximation, which we also discuss using coupled approach. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction We are concerned with the numerical solution of three-dimensional non-linear biharmonic partial differential equations of the form er 4 uðx; y; zÞ e @ 4 u @x 4 þ @ 4 u @y 4 þ @ 4 u @z 4 þ 2 @ 4 u @x 2 @y 2 þ @ 4 u @x 2 @z 2 þ @ 4 u @y 2 @z 2  !  ! ¼ f ðx; y; z; u; u x ; u y ; u z ; r 2 u; r 2 u x ; r 2 u y ; r 2 u z Þ; 0 < x; y; z < 1; ð1Þ where 0 < e 6 1, ðx; y; zÞ2 X ¼ fðx; y; zÞj0 < x; y; z < 1g with boundary @X and r 2 uðx; y; zÞ @ 2 u @x 2 þ @ 2 u @y 2 þ @ 2 u @z 2 represents the three-dimensional Laplacian of the function uðx; y; zÞ. We assume that the solution uðx; y; zÞ is smooth enough to maintain the order and accuracy of the scheme as high as possible under consideration. The Dirichlet boundary conditions are given by u ¼ aðx; y; zÞ; @ 2 u @n 2 ¼ bðx; y; zÞ; ðx; y; zÞ2 @X: ð2Þ The differential equations (1) alongwith the boundary conditions (2) are known as the second kind biharmonic problems, whereas the differential equations (1) alongwith the values of u and ð@u=@nÞ prescribed explicitly on the boundary, are known as the first kind biharmonic problems. 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.09.052 * Corresponding author. E-mail addresses: rmohanty@maths.du.ac.in, mohantyranjan@hotmail.com (R.K. Mohanty). Applied Mathematics and Computation 215 (2009) 3036–3044 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc