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A new coupled approach high accuracy numerical method for the solution of 3D non-linear biharmonic equations. (English) Zbl 1180.65137
Summary: We derive a new fourth order finite difference approximation based on arithmetic 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 $u\left(x,y,z\right)$ and its Laplacian ${\nabla }^{2}u$ 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.
##### MSC:
 65N06 Finite difference methods (BVP of PDE) 35J66 Nonlinear boundary value problems for nonlinear elliptic equations 65N15 Error bounds (BVP of PDE)