A new high-order nine-point stencil, based on integrated-RBF approximations, for the first biharmonic equation

Abstract

This paper is concerned with the development of a new compact 9-point stencil, based on integrated-radial-basis-function (IRBF) approximations, for the discretisation of the first biharmonic equation in two dimensions. Derivatives of not only the first order but also the second order and higher are included in the approximations on the stencil. These nodal derivative values, except for the boundary values of the derivative, are directly derived from nodal variable values along the grid lines rather than from the biharmonic equation, and they are updated through iteration. With these features, the double boundary conditions are imposed in a proper way. The biharmonic equation is enforced at grid points near the boundary without any special treatments. More importantly, they enable the IRBF solution to be highly accurate and not influenced by the RBF width. There is no need for searching the optimal value of the RBF width. The proposed stencil can be used to solve the biharmonic problem defined on a rectangular/non-rectangular domain. A fast convergence rate with respect to grid refinement (up to ten) is achieved.