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.

本文关注基于集成径向基函数 (IRBF) 近似的新型紧凑型 9 点模板的开发，用于在二维上离散第一个双调和方程。模板上的近似值不仅包括一阶导数，还包括二阶和更高阶的导数。这些节点导数值，除了导数的边界值，直接从沿网格线的节点变量值导出，而不是从双调和方程导出，并且它们通过迭代更新。有了这些特征，双边界条件就以适当的方式施加了。双调和方程在边界附近的网格点处强制执行，无需任何特殊处理。更重要的是，它们使 IRBF 解决方案非常准确，并且不受 RBF 宽度的影响。无需搜索RBF宽度的最佳值。所提出的模板可用于解决在矩形/非矩形域上定义的双谐波问题。实现了关于网格细化（最多十个）的快速收敛速度。