A compact discretization of O(h4) for two-dimensional nonlinear triharmonic equations

We report a new finite-difference approximation of O(h4) for two-dimensional nonlinear triharmonic partial differential equations on a nine-point compact stencil where the values of u, ∂ 2u/∂n2 and ∂ 4u/∂n4 are prescribed on the boundary. In this method, there is no need to discretize the derivative boundary conditions. The Laplacian and the biharmonic of the solution are obtained as a by-product of the method. We require only a system of three equations to obtain the solution. We compare the advantages and implementation of the proposed method with the corresponding central difference approximations of O(h2) in the context of iterative methods. Numerical results are given to verify the fourth-order convergence rate of the method.