A new coupled approach high accuracy numerical method for the solution of 3D non-linear biharmonic equations

• Published in: Applied mathematics and computation 2009-12, Vol.215 (8), p.3036-3044
• Main Authors: Khattar, Dinesh, Singh, Swarn, Mohanty, R.K.
• Format: Article
• Language: English
• Subjects: Arithmetic average discretization; Compact approximation; Exact sciences and technology; Finite differences; High accuracy; Laplacian; Mathematical analysis; Mathematics; Maximum absolute errors; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; Partial differential equations; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use; Three-dimensional non-linear biharmonic equations
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2009.09.052

In this paper, 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 ( x , y , z ) and its Laplacian ∇ 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.