Computing high-order derivatives in compact integrated-RBF stencils

Abstract

In Mai-Duy and Strunin (Mai-Duy and Strunin, 2021), it was shown that the inclusion of nodal values of high-order derivatives in compact local integrated-radial-basis-function (IRBF) stencils results in a significant improvement in the solution accuracy. The purpose of this work is to examine in detail the numerical performance of several approximation schemes based on one-dimensional IRBFs for computing high-order derivatives along the grid lines. The extended precision floating point arithmetic is utilised to achieve a high level of accuracy, and the efficiencies of the approximation schemes are improved by employing overlapping domain decomposition and mixed-precision calculations. In solving partial differential equations (PDEs), the proposed 1D-IRBFs are implemented using the RBF widths that are fixed and vary with grid refinement. A simple framework is presented to cover the two RBF width cases, and a numerical analysis is carried out for differential problems with slow and rapid variations in their solutions. In solving the convection–diffusion equations, the proposed 1D-IRBFs are also incorporated into the upwind schemes for effectively simulating highly-nonlinear flows. Numerical results show that high rates of convergence with respect to grid refinement are achieved with both fixed and variable widths.