New approximations for one-dimensional 3-point and two-dimensional 5-point compact integrated RBF stencils

Abstract

This paper presents some new compact approximation stencils based on integrated radial basis functions (IRBFs) for numerically solving second-order elliptic differential problems on Cartesian grids. Higher-order IRBF schemes are employed to approximate the field/dependent variable. The IRBF approximations in each direction are based on 3 points and constructed independently, where derivatives of the second, third, fourth, fifth and sixth orders along the grid line are enforced at the two end-points. The imposed nodal derivative values are simply acquired through a Picard-type iteration scheme. The stencil is made up of 3 points and 5 points for 1D and 2D discretisations, respectively. Numerical results show that the proposed stencils yield a high rate of convergence with respect to grid refinement, e.g. up to the 13th order for 1D problems and to the 9th order for 2D problems.