Several compact local stencils based on integrated RBFs for fourth-order ODEs and PDEs

Abstract

In this paper, new compact local stencils based on integrated radial basis functions (IRBFs) for solving fourth-order ordinary differential equations (ODEs) and partial differential equations (PDEs) are presented. Five types of compact stencils - 3-node and 5-node for 1D problems and 5×5-node, 13-node and 3×3 -node for 2D problems - are implemented. In the case of 3-node stencil and 3×3-node stencil, nodal values of the first derivative(s) of the field variable are treated as additional unknowns (i.e. 2 unknowns per node for 3-node stencil and 3 unknowns per node for 3×3-node stencil). The integration constants arising from the construction of IRBFs are exploited to incorporate into the local IRBF approximations (i) values of the governing equation (GE) at selected nodes for the case of 5-, 5×5- and 13-node stencils, and (ii) not only nodal values of the governing equation but also nodal values of the first derivative(s) for the case of 3-node stencil and 3×3-node stencil. There are no special treatments required for grid nodes near the boundary for 3-node stencil and 3×3-node stencil. The proposed stencils, which lead to sparse system matrices, are numerically verified through the solution of several test problems.